id sid tid token lemma pos cana-1458 1 1 communications communication NOUN cana-1458 1 2 on on ADP cana-1458 1 3 applied apply VERB cana-1458 1 4 nonlinear nonlinear ADJ cana-1458 1 5 analysis analysis NOUN cana-1458 1 6 issn issn NOUN cana-1458 1 7 : : PUNCT cana-1458 1 8 1074 1074 NUM cana-1458 1 9 - - PUNCT cana-1458 1 10 133x 133x NUM cana-1458 1 11 vol vol NOUN cana-1458 1 12 31 31 NUM cana-1458 1 13 no no NOUN cana-1458 1 14 . . PUNCT cana-1458 2 1 8s 8s PROPN cana-1458 2 2 ( ( PUNCT cana-1458 2 3 2024 2024 NUM cana-1458 2 4 ) ) PUNCT cana-1458 2 5 87 87 NUM cana-1458 2 6 https://internationalpubls.com https://internationalpubls.com X cana-1458 2 7 novel novel ADJ cana-1458 2 8 picture picture NOUN cana-1458 2 9 of of ADP cana-1458 2 10 topological topological ADJ cana-1458 2 11 spaces space NOUN cana-1458 2 12 in in ADP cana-1458 2 13 the the DET cana-1458 2 14 frame frame NOUN cana-1458 2 15 of of ADP cana-1458 2 16 fuzzy fuzzy ADJ cana-1458 2 17 soft soft ADJ cana-1458 2 18 sequential sequential ADJ cana-1458 2 19 sets set NOUN cana-1458 2 20 * * PUNCT cana-1458 2 21 k.japhia k.japhia ADJ cana-1458 2 22 tino tino ADJ cana-1458 2 23 mercy1 mercy1 PROPN cana-1458 2 24 , , PUNCT cana-1458 2 25 k.bageerathi2 k.bageerathi2 ADJ cana-1458 2 26 1 1 NUM cana-1458 2 27 research research NOUN cana-1458 2 28 scholar scholar NOUN cana-1458 2 29 ( ( PUNCT cana-1458 2 30 registration registration NOUN cana-1458 2 31 number number NOUN cana-1458 2 32 : : PUNCT cana-1458 2 33 22122022092006 22122022092006 NUM cana-1458 2 34 ) ) PUNCT cana-1458 2 35 , , PUNCT cana-1458 2 36 pg pg NOUN cana-1458 2 37 and and CCONJ cana-1458 2 38 research research PROPN cana-1458 2 39 department department PROPN cana-1458 2 40 of of ADP cana-1458 2 41 mathematics mathematic NOUN cana-1458 2 42 , , PUNCT cana-1458 2 43 aditanar aditanar PROPN cana-1458 2 44 college college PROPN cana-1458 2 45 of of ADP cana-1458 2 46 arts art NOUN cana-1458 2 47 and and CCONJ cana-1458 2 48 science science NOUN cana-1458 2 49 , , PUNCT cana-1458 2 50 tiruchendur tiruchendur PROPN cana-1458 2 51 , , PUNCT cana-1458 2 52 affiliated affiliate VERB cana-1458 2 53 to to ADP cana-1458 2 54 manonmaniam manonmaniam PROPN cana-1458 2 55 sundaranar sundaranar PROPN cana-1458 2 56 university university PROPN cana-1458 2 57 , , PUNCT cana-1458 2 58 abishekapatti abishekapatti PROPN cana-1458 2 59 , , PUNCT cana-1458 2 60 tirunelveli-627 tirunelveli-627 NOUN cana-1458 2 61 012 012 NOUN cana-1458 2 62 . . PROPN cana-1458 2 63 2 2 NUM cana-1458 2 64 assistant assistant NOUN cana-1458 2 65 professor professor NOUN cana-1458 2 66 , , PUNCT cana-1458 2 67 pg pg NOUN cana-1458 2 68 and and CCONJ cana-1458 2 69 research research PROPN cana-1458 2 70 department department PROPN cana-1458 2 71 of of ADP cana-1458 2 72 mathematics mathematic NOUN cana-1458 2 73 , , PUNCT cana-1458 2 74 aditanar aditanar PROPN cana-1458 2 75 college college PROPN cana-1458 2 76 of of ADP cana-1458 2 77 arts art NOUN cana-1458 2 78 and and CCONJ cana-1458 2 79 science science NOUN cana-1458 2 80 , , PUNCT cana-1458 2 81 tiruchendur tiruchendur NOUN cana-1458 2 82 . . PUNCT cana-1458 3 1 e e X cana-1458 3 2 - - NOUN cana-1458 3 3 mail mail NOUN cana-1458 3 4 i i NOUN cana-1458 3 5 d d NOUN cana-1458 3 6 : : PUNCT cana-1458 3 7 1jaffy306@gmail.com,2bageerathi14@gmail.com 1jaffy306@gmail.com,2bageerathi14@gmail.com NUM cana-1458 3 8 article article NOUN cana-1458 3 9 history history NOUN cana-1458 3 10 : : PUNCT cana-1458 3 11 received receive VERB cana-1458 3 12 : : PUNCT cana-1458 3 13 20 20 NUM cana-1458 3 14 - - PUNCT cana-1458 3 15 04 04 NUM cana-1458 3 16 - - PUNCT cana-1458 3 17 2024 2024 NUM cana-1458 3 18 revised revise VERB cana-1458 3 19 : : PUNCT cana-1458 3 20 10 10 NUM cana-1458 3 21 - - SYM cana-1458 3 22 06 06 NUM cana-1458 3 23 - - PUNCT cana-1458 3 24 2024 2024 NUM cana-1458 3 25 accepted accept VERB cana-1458 3 26 : : PUNCT cana-1458 3 27 24 24 NUM cana-1458 3 28 - - PUNCT cana-1458 3 29 06 06 NUM cana-1458 3 30 - - PUNCT cana-1458 3 31 2024 2024 NUM cana-1458 3 32 abstract abstract NOUN cana-1458 3 33 : : PUNCT cana-1458 3 34 the the DET cana-1458 3 35 goal goal NOUN cana-1458 3 36 of of ADP cana-1458 3 37 this this DET cana-1458 3 38 paper paper NOUN cana-1458 3 39 is be AUX cana-1458 3 40 to to PART cana-1458 3 41 introduce introduce VERB cana-1458 3 42 fuzzy fuzzy ADJ cana-1458 3 43 soft soft ADJ cana-1458 3 44 sequential sequential ADJ cana-1458 3 45 topological topological ADJ cana-1458 3 46 spaces space NOUN cana-1458 3 47 over over ADP cana-1458 3 48 the the DET cana-1458 3 49 universal universal ADJ cana-1458 3 50 set set NOUN cana-1458 3 51 via via ADP cana-1458 3 52 fuzzy fuzzy ADJ cana-1458 3 53 soft soft ADJ cana-1458 3 54 sequential sequential ADJ cana-1458 3 55 set set VERB cana-1458 3 56 by by ADP cana-1458 3 57 merging merge VERB cana-1458 3 58 the the DET cana-1458 3 59 concept concept NOUN cana-1458 3 60 of of ADP cana-1458 3 61 fuzzy fuzzy ADJ cana-1458 3 62 soft soft ADJ cana-1458 3 63 set set NOUN cana-1458 3 64 and and CCONJ cana-1458 3 65 sequential sequential ADJ cana-1458 3 66 set set NOUN cana-1458 3 67 . . PUNCT cana-1458 4 1 we we PRON cana-1458 4 2 provide provide VERB cana-1458 4 3 a a DET cana-1458 4 4 comprehensive comprehensive ADJ cana-1458 4 5 study study NOUN cana-1458 4 6 of of ADP cana-1458 4 7 their their PRON cana-1458 4 8 properties property NOUN cana-1458 4 9 and and CCONJ cana-1458 4 10 explore explore VERB cana-1458 4 11 a a DET cana-1458 4 12 relationship relationship NOUN cana-1458 4 13 between between ADP cana-1458 4 14 fuzzy fuzzy ADJ cana-1458 4 15 soft soft ADJ cana-1458 4 16 topology topology NOUN cana-1458 4 17 and and CCONJ cana-1458 4 18 fuzzy fuzzy ADJ cana-1458 4 19 soft soft ADJ cana-1458 4 20 sequential sequential ADJ cana-1458 4 21 topology topology NOUN cana-1458 4 22 . . PUNCT cana-1458 5 1 we we PRON cana-1458 5 2 offers offer VERB cana-1458 5 3 a a DET cana-1458 5 4 new new ADJ cana-1458 5 5 outlook outlook NOUN cana-1458 5 6 on on ADP cana-1458 5 7 the the DET cana-1458 5 8 theory theory NOUN cana-1458 5 9 of of ADP cana-1458 5 10 fuzzy fuzzy ADJ cana-1458 5 11 soft soft ADJ cana-1458 5 12 sequential sequential ADJ cana-1458 5 13 neighborhood neighborhood NOUN cana-1458 5 14 of of ADP cana-1458 5 15 a a DET cana-1458 5 16 fuzzy fuzzy ADJ cana-1458 5 17 soft soft ADJ cana-1458 5 18 sequential sequential ADJ cana-1458 5 19 set set NOUN cana-1458 5 20 and and CCONJ cana-1458 5 21 we we PRON cana-1458 5 22 investigate investigate VERB cana-1458 5 23 the the DET cana-1458 5 24 quasi quasi ADJ cana-1458 5 25 - - ADJ cana-1458 5 26 coincident coincident ADJ cana-1458 5 27 relation relation NOUN cana-1458 5 28 . . PUNCT cana-1458 6 1 keywords keyword NOUN cana-1458 6 2 : : PUNCT cana-1458 6 3 fuzzy fuzzy ADJ cana-1458 6 4 soft soft ADJ cana-1458 6 5 set set NOUN cana-1458 6 6 , , PUNCT cana-1458 6 7 fuzzy fuzzy ADJ cana-1458 6 8 soft soft ADJ cana-1458 6 9 sequential sequential ADJ cana-1458 6 10 set set NOUN cana-1458 6 11 , , PUNCT cana-1458 6 12 fuzzy fuzzy ADJ cana-1458 6 13 soft soft ADJ cana-1458 6 14 sequential sequential ADJ cana-1458 6 15 topological topological ADJ cana-1458 6 16 spaces space NOUN cana-1458 6 17 , , PUNCT cana-1458 6 18 fuzzy fuzzy ADJ cana-1458 6 19 soft soft ADJ cana-1458 6 20 sequential sequential ADJ cana-1458 6 21 neighborhood neighborhood NOUN cana-1458 6 22 system system NOUN cana-1458 6 23 , , PUNCT cana-1458 6 24 fuzzy fuzzy ADJ cana-1458 6 25 soft soft ADJ cana-1458 6 26 sequential sequential ADJ cana-1458 6 27 quasi quasi NOUN cana-1458 6 28 - - ADJ cana-1458 6 29 coincident coincident ADJ cana-1458 6 30 1 1 NUM cana-1458 6 31 . . PUNCT cana-1458 6 32 introduction introduction NOUN cana-1458 6 33 the the DET cana-1458 6 34 concept concept NOUN cana-1458 6 35 of of ADP cana-1458 6 36 fuzzy fuzzy ADJ cana-1458 6 37 sets set NOUN cana-1458 6 38 by by ADP cana-1458 6 39 defining define VERB cana-1458 6 40 them they PRON cana-1458 6 41 in in ADP cana-1458 6 42 terms term NOUN cana-1458 6 43 of of ADP cana-1458 6 44 mappings mapping NOUN cana-1458 6 45 from from ADP cana-1458 6 46 a a DET cana-1458 6 47 set set NOUN cana-1458 6 48 into into ADP cana-1458 6 49 the the DET cana-1458 6 50 unit unit NOUN cana-1458 6 51 interval interval NOUN cana-1458 6 52 on on ADP cana-1458 6 53 the the DET cana-1458 6 54 real real ADJ cana-1458 6 55 line line NOUN cana-1458 6 56 was be AUX cana-1458 6 57 proposed propose VERB cana-1458 6 58 by by ADP cana-1458 6 59 l.a.zadeh l.a.zadeh NOUN cana-1458 6 60 [ [ X cana-1458 6 61 18 18 NUM cana-1458 6 62 ] ] PUNCT cana-1458 6 63 in in ADP cana-1458 6 64 1965 1965 NUM cana-1458 6 65 , , PUNCT cana-1458 6 66 which which PRON cana-1458 6 67 is be AUX cana-1458 6 68 dealing deal VERB cana-1458 6 69 with with ADP cana-1458 6 70 uncertainty uncertainty NOUN cana-1458 6 71 and and CCONJ cana-1458 6 72 representing represent VERB cana-1458 6 73 vague vague ADJ cana-1458 6 74 concepts concept NOUN cana-1458 6 75 . . PUNCT cana-1458 7 1 in in ADP cana-1458 7 2 1968 1968 NUM cana-1458 7 3 , , PUNCT cana-1458 7 4 c. c. PROPN cana-1458 7 5 l. l. PROPN cana-1458 7 6 chang chang PROPN cana-1458 8 1 [ [ X cana-1458 8 2 4 4 X cana-1458 8 3 ] ] PUNCT cana-1458 8 4 initiated initiate VERB cana-1458 8 5 fuzzy fuzzy ADJ cana-1458 8 6 topological topological ADJ cana-1458 8 7 spaces space NOUN cana-1458 8 8 as as ADP cana-1458 8 9 an an DET cana-1458 8 10 extension extension NOUN cana-1458 8 11 to to ADP cana-1458 8 12 classical classical ADJ cana-1458 8 13 topological topological ADJ cana-1458 8 14 spaces space NOUN cana-1458 8 15 . . PUNCT cana-1458 9 1 in in ADP cana-1458 9 2 1999 1999 NUM cana-1458 9 3 , , PUNCT cana-1458 9 4 molodtsov molodtsov NOUN cana-1458 9 5 [ [ X cana-1458 9 6 11 11 NUM cana-1458 9 7 ] ] PUNCT cana-1458 9 8 introduced introduce VERB cana-1458 9 9 the the DET cana-1458 9 10 concept concept NOUN cana-1458 9 11 of of ADP cana-1458 9 12 soft soft ADJ cana-1458 9 13 set set NOUN cana-1458 9 14 theory theory NOUN cana-1458 9 15 , , PUNCT cana-1458 9 16 which which PRON cana-1458 9 17 provides provide VERB cana-1458 9 18 a a DET cana-1458 9 19 new new ADJ cana-1458 9 20 mathematical mathematical ADJ cana-1458 9 21 theory theory NOUN cana-1458 9 22 for for ADP cana-1458 9 23 dealing deal VERB cana-1458 9 24 with with ADP cana-1458 9 25 uncertainty uncertainty NOUN cana-1458 9 26 . . PUNCT cana-1458 10 1 in in ADP cana-1458 10 2 2003 2003 NUM cana-1458 10 3 , , PUNCT cana-1458 10 4 maji maji PROPN cana-1458 10 5 et et PROPN cana-1458 10 6 al al PROPN cana-1458 11 1 [ [ X cana-1458 11 2 10 10 NUM cana-1458 11 3 ] ] PUNCT cana-1458 11 4 defined define VERB cana-1458 11 5 and and CCONJ cana-1458 11 6 studied study VERB cana-1458 11 7 the the DET cana-1458 11 8 theory theory NOUN cana-1458 11 9 of of ADP cana-1458 11 10 soft soft ADJ cana-1458 11 11 sets set NOUN cana-1458 11 12 which which PRON cana-1458 11 13 is be AUX cana-1458 11 14 used use VERB cana-1458 11 15 to to PART cana-1458 11 16 construct construct VERB cana-1458 11 17 new new ADJ cana-1458 11 18 soft soft ADJ cana-1458 11 19 sets set NOUN cana-1458 11 20 from from ADP cana-1458 11 21 the the DET cana-1458 11 22 given give VERB cana-1458 11 23 soft soft ADJ cana-1458 11 24 sets set NOUN cana-1458 11 25 . . PUNCT cana-1458 12 1 in in ADP cana-1458 12 2 2011 2011 NUM cana-1458 12 3 , , PUNCT cana-1458 12 4 shabir shabir NOUN cana-1458 12 5 and and CCONJ cana-1458 12 6 naz naz PROPN cana-1458 13 1 [ [ X cana-1458 13 2 14 14 NUM cana-1458 13 3 ] ] PUNCT cana-1458 13 4 introduced introduce VERB cana-1458 13 5 soft soft ADJ cana-1458 13 6 topological topological ADJ cana-1458 13 7 spaces space NOUN cana-1458 13 8 which which PRON cana-1458 13 9 are be AUX cana-1458 13 10 defined define VERB cana-1458 13 11 over over ADP cana-1458 13 12 an an DET cana-1458 13 13 initial initial ADJ cana-1458 13 14 universe universe NOUN cana-1458 13 15 with with ADP cana-1458 13 16 a a DET cana-1458 13 17 fixed fix VERB cana-1458 13 18 set set NOUN cana-1458 13 19 of of ADP cana-1458 13 20 parameters parameter NOUN cana-1458 13 21 . . PUNCT cana-1458 14 1 further far ADV cana-1458 14 2 in in ADP cana-1458 14 3 the the DET cana-1458 14 4 same same ADJ cana-1458 14 5 year year NOUN cana-1458 14 6 , , PUNCT cana-1458 14 7 hussain hussain PROPN cana-1458 14 8 and and CCONJ cana-1458 14 9 ahmad ahmad PROPN cana-1458 14 10 [ [ X cana-1458 14 11 6 6 NUM cana-1458 14 12 ] ] PUNCT cana-1458 14 13 extended extend VERB cana-1458 14 14 the the DET cana-1458 14 15 soft soft ADJ cana-1458 14 16 topology topology NOUN cana-1458 14 17 and and CCONJ cana-1458 14 18 strengthen strengthen VERB cana-1458 14 19 the the DET cana-1458 14 20 foundations foundation NOUN cana-1458 14 21 of of ADP cana-1458 14 22 the the DET cana-1458 14 23 theory theory NOUN cana-1458 14 24 of of ADP cana-1458 14 25 soft soft ADJ cana-1458 14 26 topological topological ADJ cana-1458 14 27 spaces space NOUN cana-1458 14 28 . . PUNCT cana-1458 15 1 also also ADV cana-1458 15 2 cagman cagman ADJ cana-1458 15 3 , , PUNCT cana-1458 15 4 enginoglu enginoglu NOUN cana-1458 15 5 and and CCONJ cana-1458 15 6 citak citak NOUN cana-1458 16 1 [ [ X cana-1458 16 2 3 3 NUM cana-1458 16 3 ] ] PUNCT cana-1458 16 4 investigated investigate VERB cana-1458 16 5 soft soft ADJ cana-1458 16 6 topology topology NOUN cana-1458 16 7 and and CCONJ cana-1458 16 8 its its PRON cana-1458 16 9 properties property NOUN cana-1458 16 10 . . PUNCT cana-1458 17 1 in in ADP cana-1458 17 2 recent recent ADJ cana-1458 17 3 years year NOUN cana-1458 17 4 , , PUNCT cana-1458 17 5 the the DET cana-1458 17 6 researchers researcher NOUN cana-1458 17 7 have have AUX cana-1458 17 8 contributed contribute VERB cana-1458 17 9 a a DET cana-1458 17 10 lot lot NOUN cana-1458 17 11 towards towards ADP cana-1458 17 12 fuzzification fuzzification NOUN cana-1458 17 13 of of ADP cana-1458 17 14 soft soft ADJ cana-1458 17 15 set set NOUN cana-1458 17 16 theory theory NOUN cana-1458 17 17 . . PUNCT cana-1458 18 1 in in ADP cana-1458 18 2 2001 2001 NUM cana-1458 18 3 , , PUNCT cana-1458 18 4 maji maji PROPN cana-1458 18 5 et et PROPN cana-1458 18 6 al al PROPN cana-1458 19 1 [ [ X cana-1458 19 2 9 9 NUM cana-1458 19 3 ] ] PUNCT cana-1458 19 4 proposed propose VERB cana-1458 19 5 the the DET cana-1458 19 6 concept concept NOUN cana-1458 19 7 of of ADP cana-1458 19 8 fuzzy fuzzy ADJ cana-1458 19 9 soft soft ADJ cana-1458 19 10 set set NOUN cana-1458 19 11 which which PRON cana-1458 19 12 is be AUX cana-1458 19 13 a a DET cana-1458 19 14 new new ADJ cana-1458 19 15 mathematical mathematical ADJ cana-1458 19 16 approach approach NOUN cana-1458 19 17 to to ADP cana-1458 19 18 vagueness vagueness NOUN cana-1458 19 19 by by ADP cana-1458 19 20 involving involve VERB cana-1458 19 21 the the DET cana-1458 19 22 ideas idea NOUN cana-1458 19 23 of of ADP cana-1458 19 24 both both CCONJ cana-1458 19 25 fuzzy fuzzy ADJ cana-1458 19 26 sets set NOUN cana-1458 19 27 and and CCONJ cana-1458 19 28 soft soft ADJ cana-1458 19 29 sets set NOUN cana-1458 19 30 . . PUNCT cana-1458 20 1 fuzzy fuzzy ADJ cana-1458 20 2 soft soft ADJ cana-1458 20 3 sets set NOUN cana-1458 20 4 was be AUX cana-1458 20 5 further far ADV cana-1458 20 6 revised revise VERB cana-1458 20 7 and and CCONJ cana-1458 20 8 improved improve VERB cana-1458 20 9 by by ADP cana-1458 20 10 ahmad ahmad NOUN cana-1458 20 11 and and CCONJ cana-1458 20 12 kharal kharal ADJ cana-1458 20 13 [ [ X cana-1458 20 14 1 1 NUM cana-1458 20 15 ] ] PUNCT cana-1458 20 16 in in ADP cana-1458 20 17 the the DET cana-1458 20 18 year year NOUN cana-1458 20 19 2009 2009 NUM cana-1458 20 20 . . PUNCT cana-1458 21 1 in in ADP cana-1458 21 2 2011 2011 NUM cana-1458 21 3 , , PUNCT cana-1458 21 4 tanay tanay PROPN cana-1458 21 5 and and CCONJ cana-1458 21 6 kandemir kandemir X cana-1458 22 1 [ [ X cana-1458 22 2 16 16 NUM cana-1458 22 3 ] ] PUNCT cana-1458 22 4 introduced introduce VERB cana-1458 22 5 the the DET cana-1458 22 6 topological topological ADJ cana-1458 22 7 structure structure NOUN cana-1458 22 8 of of ADP cana-1458 22 9 fuzzy fuzzy ADJ cana-1458 22 10 soft soft ADJ cana-1458 22 11 sets set NOUN cana-1458 22 12 and and CCONJ cana-1458 22 13 studied study VERB cana-1458 22 14 some some PRON cana-1458 22 15 of of ADP cana-1458 22 16 its its PRON cana-1458 22 17 structural structural ADJ cana-1458 22 18 properties property NOUN cana-1458 22 19 . . PUNCT cana-1458 23 1 based base VERB cana-1458 23 2 on on ADP cana-1458 23 3 this this DET cana-1458 23 4 theory theory NOUN cana-1458 23 5 many many ADJ cana-1458 23 6 researchers researcher NOUN cana-1458 23 7 like like ADP cana-1458 23 8 roy roy PROPN cana-1458 23 9 and and CCONJ cana-1458 23 10 samanta samanta PROPN cana-1458 24 1 [ [ X cana-1458 24 2 12 12 NUM cana-1458 24 3 ] ] PUNCT cana-1458 24 4 in in ADP cana-1458 24 5 the the DET cana-1458 24 6 year year NOUN cana-1458 24 7 2012 2012 NUM cana-1458 24 8 , , PUNCT cana-1458 24 9 atmaca atmaca NOUN cana-1458 24 10 and and CCONJ cana-1458 24 11 zorlutuna zorlutuna PROPN cana-1458 25 1 [ [ X cana-1458 25 2 13 13 NUM cana-1458 25 3 ] ] PUNCT cana-1458 25 4 in in ADP cana-1458 25 5 the the DET cana-1458 25 6 year year NOUN cana-1458 25 7 2013 2013 NUM cana-1458 25 8 , , PUNCT cana-1458 25 9 mahanta mahanta ADJ cana-1458 25 10 and and CCONJ cana-1458 25 11 das das PROPN cana-1458 26 1 [ [ X cana-1458 26 2 8 8 NUM cana-1458 26 3 ] ] PUNCT cana-1458 26 4 in in ADP cana-1458 26 5 the the DET cana-1458 26 6 year year NOUN cana-1458 26 7 2018 2018 NUM cana-1458 26 8 studied study VERB cana-1458 26 9 and and CCONJ cana-1458 26 10 investigated investigate VERB cana-1458 26 11 many many ADJ cana-1458 26 12 properties property NOUN cana-1458 26 13 in in ADP cana-1458 26 14 fuzzy fuzzy ADJ cana-1458 26 15 soft soft ADJ cana-1458 26 16 topology topology NOUN cana-1458 26 17 . . PUNCT cana-1458 27 1 in in ADP cana-1458 27 2 2002 2002 NUM cana-1458 27 3 , , PUNCT cana-1458 27 4 bose bose NOUN cana-1458 27 5 and and CCONJ cana-1458 27 6 indrajit indrajit ADJ cana-1458 27 7 lahiri lahiri NOUN cana-1458 28 1 [ [ X cana-1458 28 2 2 2 NUM cana-1458 28 3 ] ] PUNCT cana-1458 28 4 communications communication NOUN cana-1458 28 5 on on ADP cana-1458 28 6 applied apply VERB cana-1458 28 7 nonlinear nonlinear ADJ cana-1458 28 8 analysis analysis NOUN cana-1458 28 9 issn issn NOUN cana-1458 28 10 : : PUNCT cana-1458 28 11 1074 1074 NUM cana-1458 28 12 - - PUNCT cana-1458 28 13 133x 133x NUM cana-1458 28 14 vol vol NOUN cana-1458 28 15 31 31 NUM cana-1458 28 16 no no NOUN cana-1458 28 17 . . PUNCT cana-1458 29 1 8s 8s PROPN cana-1458 29 2 ( ( PUNCT cana-1458 29 3 2024 2024 NUM cana-1458 29 4 ) ) PUNCT cana-1458 29 5 88 88 NUM cana-1458 29 6 https://internationalpubls.com https://internationalpubls.com X cana-1458 29 7 introduced introduce VERB cana-1458 29 8 the the DET cana-1458 29 9 concept concept NOUN cana-1458 29 10 of of ADP cana-1458 29 11 sequential sequential ADJ cana-1458 29 12 topological topological ADJ cana-1458 29 13 spaces space NOUN cana-1458 29 14 . . PUNCT cana-1458 30 1 he he PRON cana-1458 30 2 defined define VERB cana-1458 30 3 any any DET cana-1458 30 4 sequence sequence NOUN cana-1458 30 5 of of ADP cana-1458 30 6 subsets subset NOUN cana-1458 30 7 of of ADP cana-1458 30 8 a a DET cana-1458 30 9 non non ADJ cana-1458 30 10 void void NOUN cana-1458 30 11 set set NOUN cana-1458 30 12 is be AUX cana-1458 30 13 called call VERB cana-1458 30 14 a a DET cana-1458 30 15 sequential sequential ADJ cana-1458 30 16 set set NOUN cana-1458 30 17 . . PUNCT cana-1458 31 1 to to PART cana-1458 31 2 go go VERB cana-1458 31 3 along along ADP cana-1458 31 4 this this DET cana-1458 31 5 line line NOUN cana-1458 31 6 of of ADP cana-1458 31 7 research research NOUN cana-1458 31 8 , , PUNCT cana-1458 31 9 we we PRON cana-1458 31 10 introduce introduce VERB cana-1458 31 11 sequence sequence NOUN cana-1458 31 12 of of ADP cana-1458 31 13 fuzzy fuzzy ADJ cana-1458 31 14 soft soft ADJ cana-1458 31 15 set set NOUN cana-1458 31 16 and and CCONJ cana-1458 31 17 named name VERB cana-1458 31 18 as as ADP cana-1458 31 19 fuzzy fuzzy ADJ cana-1458 31 20 soft soft ADJ cana-1458 31 21 sequential sequential ADJ cana-1458 31 22 set set NOUN cana-1458 31 23 . . PUNCT cana-1458 32 1 in in ADP cana-1458 32 2 this this DET cana-1458 32 3 paper paper NOUN cana-1458 32 4 , , PUNCT cana-1458 32 5 we we PRON cana-1458 32 6 extend extend VERB cana-1458 32 7 the the DET cana-1458 32 8 concept concept NOUN cana-1458 32 9 of of ADP cana-1458 32 10 topological topological ADJ cana-1458 32 11 space space NOUN cana-1458 32 12 namely namely ADV cana-1458 32 13 fuzzy fuzzy ADJ cana-1458 32 14 soft soft ADJ cana-1458 32 15 sequential sequential ADJ cana-1458 32 16 topological topological ADJ cana-1458 32 17 space space NOUN cana-1458 32 18 and and CCONJ cana-1458 32 19 analyze analyze VERB cana-1458 32 20 their their PRON cana-1458 32 21 properties property NOUN cana-1458 32 22 . . PUNCT cana-1458 33 1 2 2 X cana-1458 33 2 . . X cana-1458 33 3 preliminaries preliminary NOUN cana-1458 33 4 definition definition NOUN cana-1458 33 5 2.1 2.1 NUM cana-1458 33 6 [ [ SYM cana-1458 33 7 18 18 NUM cana-1458 33 8 ] ] PUNCT cana-1458 33 9 a a DET cana-1458 33 10 fuzzy fuzzy ADJ cana-1458 33 11 set set VERB cana-1458 33 12 𝑋 𝑋 NOUN cana-1458 33 13 over over ADP cana-1458 33 14 a a DET cana-1458 33 15 universal universal ADJ cana-1458 33 16 set set NOUN cana-1458 33 17 π‘ˆ π‘ˆ PROPN cana-1458 33 18 is be AUX cana-1458 33 19 a a DET cana-1458 33 20 set set NOUN cana-1458 33 21 defined define VERB cana-1458 33 22 by by ADP cana-1458 33 23 a a DET cana-1458 33 24 function function NOUN cana-1458 33 25 πœ‡π‘₯ πœ‡π‘₯ AUX cana-1458 33 26 performing perform VERB cana-1458 33 27 a a DET cana-1458 33 28 mapping mapping NOUN cana-1458 33 29 πœ‡π‘₯ πœ‡π‘₯ NOUN cana-1458 33 30 ∢ ∢ NOUN cana-1458 33 31 π‘ˆ π‘ˆ PROPN cana-1458 33 32 β†’ β†’ SYM cana-1458 33 33 [ [ X cana-1458 33 34 0,1 0,1 NUM cana-1458 33 35 ] ] PUNCT cana-1458 33 36 , , PUNCT cana-1458 33 37 here here ADV cana-1458 33 38 this this DET cana-1458 33 39 πœ‡π‘₯ πœ‡π‘₯ NOUN cana-1458 33 40 is be AUX cana-1458 33 41 the the DET cana-1458 33 42 membership membership NOUN cana-1458 33 43 function function NOUN cana-1458 33 44 of of ADP cana-1458 33 45 𝑋 𝑋 PROPN cana-1458 33 46 , , PUNCT cana-1458 33 47 and and CCONJ cana-1458 33 48 the the DET cana-1458 33 49 value value NOUN cana-1458 33 50 πœ‡π‘₯(𝑒 πœ‡π‘₯(𝑒 NUM cana-1458 33 51 ) ) PUNCT cana-1458 33 52 will will AUX cana-1458 33 53 be be AUX cana-1458 33 54 the the DET cana-1458 33 55 grade grade NOUN cana-1458 33 56 of of ADP cana-1458 33 57 membership membership NOUN cana-1458 33 58 of of ADP cana-1458 33 59 u u PROPN cana-1458 33 60 ∈ ∈ PROPN cana-1458 33 61 π‘ˆ. π‘ˆ. PROPN cana-1458 33 62 definition definition NOUN cana-1458 33 63 2.2 2.2 NUM cana-1458 33 64 [ [ SYM cana-1458 33 65 11 11 NUM cana-1458 33 66 ] ] PUNCT cana-1458 33 67 let let VERB cana-1458 33 68 π‘ˆ π‘ˆ PROPN cana-1458 33 69 be be AUX cana-1458 33 70 an an DET cana-1458 33 71 initial initial ADJ cana-1458 33 72 universe universe NOUN cana-1458 33 73 set set VERB cana-1458 33 74 and and CCONJ cana-1458 33 75 𝐸 𝐸 PROPN cana-1458 33 76 be be VERB cana-1458 33 77 a a DET cana-1458 33 78 set set NOUN cana-1458 33 79 of of ADP cana-1458 33 80 parameters parameter NOUN cana-1458 33 81 . . PUNCT cana-1458 34 1 a a DET cana-1458 34 2 pair pair NOUN cana-1458 34 3 ( ( PUNCT cana-1458 34 4 𝐹 𝐹 PROPN cana-1458 34 5 , , PUNCT cana-1458 34 6 𝐸 𝐸 PROPN cana-1458 34 7 ) ) PUNCT cana-1458 34 8 is be AUX cana-1458 34 9 called call VERB cana-1458 34 10 a a DET cana-1458 34 11 soft soft ADJ cana-1458 34 12 set set NOUN cana-1458 34 13 over over ADP cana-1458 34 14 π‘ˆ π‘ˆ PROPN cana-1458 34 15 if if SCONJ cana-1458 34 16 and and CCONJ cana-1458 34 17 only only ADV cana-1458 34 18 if if SCONJ cana-1458 34 19 𝐹 𝐹 PROPN cana-1458 34 20 is be AUX cana-1458 34 21 a a DET cana-1458 34 22 mapping mapping NOUN cana-1458 34 23 of of ADP cana-1458 34 24 𝐸 𝐸 PROPN cana-1458 34 25 into into ADP cana-1458 34 26 the the DET cana-1458 34 27 set set NOUN cana-1458 34 28 of of ADP cana-1458 34 29 all all DET cana-1458 34 30 subsets subset NOUN cana-1458 34 31 of of ADP cana-1458 34 32 the the DET cana-1458 34 33 set set ADJ cana-1458 34 34 π‘ˆ. π‘ˆ. PROPN cana-1458 34 35 definition definition NOUN cana-1458 34 36 2.3 2.3 NUM cana-1458 34 37 [ [ X cana-1458 34 38 9 9 NUM cana-1458 34 39 ] ] PUNCT cana-1458 34 40 let let VERB cana-1458 34 41 π‘ˆ π‘ˆ PROPN cana-1458 34 42 be be AUX cana-1458 34 43 a a DET cana-1458 34 44 common common ADJ cana-1458 34 45 universe universe NOUN cana-1458 34 46 , , PUNCT cana-1458 34 47 𝐸 𝐸 PROPN cana-1458 34 48 be be VERB cana-1458 34 49 a a DET cana-1458 34 50 set set NOUN cana-1458 34 51 of of ADP cana-1458 34 52 parameters parameter NOUN cana-1458 34 53 and and CCONJ cana-1458 34 54 𝐴 𝐴 PROPN cana-1458 34 55 βŠ† βŠ† NUM cana-1458 34 56 𝐸. 𝐸. VERB cana-1458 34 57 a a DET cana-1458 34 58 pair pair NOUN cana-1458 34 59 ( ( PUNCT cana-1458 34 60 𝐹 𝐹 PROPN cana-1458 34 61 , , PUNCT cana-1458 34 62 𝐴 𝐴 PROPN cana-1458 34 63 ) ) PUNCT cana-1458 34 64 is be AUX cana-1458 34 65 called call VERB cana-1458 34 66 a a DET cana-1458 34 67 fuzzy fuzzy ADJ cana-1458 34 68 soft soft ADJ cana-1458 34 69 set set NOUN cana-1458 34 70 over over ADP cana-1458 34 71 π‘ˆ π‘ˆ PROPN cana-1458 34 72 , , PUNCT cana-1458 34 73 where where SCONJ cana-1458 34 74 𝐹 𝐹 PROPN cana-1458 34 75 is be AUX cana-1458 34 76 a a DET cana-1458 34 77 mapping mapping NOUN cana-1458 34 78 given give VERB cana-1458 34 79 by by ADP cana-1458 34 80 𝐹 𝐹 PROPN cana-1458 34 81 : : PUNCT cana-1458 34 82 𝐴 𝐴 PROPN cana-1458 34 83 β†’ β†’ SYM cana-1458 34 84 πœ‡π‘₯(π‘ˆ πœ‡π‘₯(π‘ˆ PROPN cana-1458 34 85 ) ) PUNCT cana-1458 34 86 , , PUNCT cana-1458 34 87 where where SCONJ cana-1458 34 88 πœ‡π‘₯(π‘ˆ πœ‡π‘₯(π‘ˆ NOUN cana-1458 34 89 ) ) PUNCT cana-1458 34 90 denote denote VERB cana-1458 34 91 the the DET cana-1458 34 92 set set NOUN cana-1458 34 93 of of ADP cana-1458 34 94 all all DET cana-1458 34 95 fuzzy fuzzy ADJ cana-1458 34 96 subsets subset NOUN cana-1458 34 97 of of ADP cana-1458 34 98 π‘ˆ. π‘ˆ. ADJ cana-1458 34 99 definition definition NOUN cana-1458 34 100 2.4 2.4 NUM cana-1458 34 101 [ [ SYM cana-1458 34 102 16 16 NUM cana-1458 34 103 ] ] PUNCT cana-1458 34 104 let let VERB cana-1458 34 105 π‘ˆ π‘ˆ PROPN cana-1458 34 106 be be AUX cana-1458 34 107 a a DET cana-1458 34 108 common common ADJ cana-1458 34 109 universe universe NOUN cana-1458 34 110 , , PUNCT cana-1458 34 111 𝐸 𝐸 PROPN cana-1458 34 112 be be VERB cana-1458 34 113 a a DET cana-1458 34 114 set set NOUN cana-1458 34 115 of of ADP cana-1458 34 116 parameters parameter NOUN cana-1458 34 117 and and CCONJ cana-1458 34 118 𝐴 𝐴 PROPN cana-1458 34 119 , , PUNCT cana-1458 34 120 𝐡 𝐡 PROPN cana-1458 34 121 βŠ† βŠ† NUM cana-1458 34 122 𝐸. 𝐸. PROPN cana-1458 34 123 let let VERB cana-1458 34 124 ( ( PUNCT cana-1458 34 125 𝐹 𝐹 PROPN cana-1458 34 126 , , PUNCT cana-1458 34 127 𝐴 𝐴 PROPN cana-1458 34 128 ) ) PUNCT cana-1458 34 129 , , PUNCT cana-1458 34 130 ( ( PUNCT cana-1458 34 131 𝐺 𝐺 NOUN cana-1458 34 132 , , PUNCT cana-1458 34 133 𝐡 𝐡 PROPN cana-1458 34 134 ) ) PUNCT cana-1458 34 135 be be VERB cana-1458 34 136 an an DET cana-1458 34 137 element element NOUN cana-1458 34 138 of of ADP cana-1458 34 139 𝐹𝑆(π‘ˆ 𝐹𝑆(π‘ˆ PROPN cana-1458 34 140 , , PUNCT cana-1458 34 141 𝐸 𝐸 PROPN cana-1458 34 142 ) ) PUNCT cana-1458 34 143 , , PUNCT cana-1458 34 144 ( ( PUNCT cana-1458 34 145 briefly briefly ADV cana-1458 34 146 , , PUNCT cana-1458 34 147 family family NOUN cana-1458 34 148 of of ADP cana-1458 34 149 all all DET cana-1458 34 150 fuzzy fuzzy ADJ cana-1458 34 151 soft soft ADJ cana-1458 34 152 subsets subset NOUN cana-1458 34 153 ) ) PUNCT cana-1458 34 154 and and CCONJ cana-1458 34 155 οΏ½ οΏ½ PROPN cana-1458 34 156 Μƒ Μƒ PROPN cana-1458 34 157 οΏ½ οΏ½ PROPN cana-1458 34 158 be be AUX cana-1458 34 159 a a DET cana-1458 34 160 subfamily subfamily NOUN cana-1458 34 161 of of ADP cana-1458 34 162 𝐹𝑆(π‘ˆ 𝐹𝑆(π‘ˆ NOUN cana-1458 34 163 , , PUNCT cana-1458 34 164 𝐸 𝐸 PROPN cana-1458 34 165 ) ) PUNCT cana-1458 34 166 . . PUNCT cana-1458 35 1 then then ADV cana-1458 35 2 οΏ½ οΏ½ PROPN cana-1458 35 3 Μƒ Μƒ PROPN cana-1458 35 4 οΏ½ οΏ½ PROPN cana-1458 35 5 is be AUX cana-1458 35 6 called call VERB cana-1458 35 7 a a DET cana-1458 35 8 fuzzy fuzzy ADJ cana-1458 35 9 soft soft ADJ cana-1458 35 10 topology topology NOUN cana-1458 35 11 on on ADP cana-1458 35 12 π‘ˆ π‘ˆ PROPN cana-1458 35 13 if if SCONJ cana-1458 35 14 the the DET cana-1458 35 15 following follow VERB cana-1458 35 16 conditions condition NOUN cana-1458 35 17 are be AUX cana-1458 35 18 satisfied satisfied ADJ cana-1458 35 19 : : PUNCT cana-1458 35 20 ( ( PUNCT cana-1458 35 21 i i NOUN cana-1458 35 22 ) ) PUNCT cana-1458 35 23 βˆ…Μƒ βˆ…Μƒ NOUN cana-1458 35 24 , , PUNCT cana-1458 35 25 π‘ˆ π‘ˆ PROPN cana-1458 35 26 ∈ ∈ PROPN cana-1458 35 27 οΏ½ οΏ½ PROPN cana-1458 35 28 Μƒ Μƒ PROPN cana-1458 35 29 οΏ½ οΏ½ PROPN cana-1458 35 30 ( ( PUNCT cana-1458 35 31 ii ii NOUN cana-1458 35 32 ) ) PUNCT cana-1458 35 33 ( ( PUNCT cana-1458 35 34 𝐹 𝐹 PROPN cana-1458 35 35 , , PUNCT cana-1458 35 36 𝐴 𝐴 PROPN cana-1458 35 37 ) ) PUNCT cana-1458 35 38 , , PUNCT cana-1458 35 39 ( ( PUNCT cana-1458 35 40 𝐺 𝐺 NOUN cana-1458 35 41 , , PUNCT cana-1458 35 42 𝐡 𝐡 PROPN cana-1458 35 43 ) ) PUNCT cana-1458 35 44 ∈ ∈ PROPN cana-1458 35 45 οΏ½ οΏ½ PROPN cana-1458 35 46 Μƒ Μƒ PROPN cana-1458 35 47 οΏ½ οΏ½ PROPN cana-1458 35 48 β‡’ β‡’ NOUN cana-1458 35 49 ( ( PUNCT cana-1458 35 50 𝐹 𝐹 PROPN cana-1458 35 51 , , PUNCT cana-1458 35 52 𝐴 𝐴 PROPN cana-1458 35 53 ) ) PUNCT cana-1458 35 54 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 35 55 ( ( PUNCT cana-1458 35 56 𝐺 𝐺 PROPN cana-1458 35 57 , , PUNCT cana-1458 35 58 𝐡 𝐡 PROPN cana-1458 35 59 ) ) PUNCT cana-1458 35 60 ∈ ∈ PROPN cana-1458 35 61 οΏ½ οΏ½ PROPN cana-1458 35 62 Μƒ Μƒ PROPN cana-1458 35 63 οΏ½ οΏ½ PROPN cana-1458 35 64 ( ( PUNCT cana-1458 35 65 iii){(𝐹 iii){(𝐹 PROPN cana-1458 35 66 , , PUNCT cana-1458 35 67 𝐴)π‘˜|π‘˜ 𝐴)π‘˜|π‘˜ NOUN cana-1458 35 68 ∈ ∈ PROPN cana-1458 35 69 𝐾 𝐾 PROPN cana-1458 35 70 } } PUNCT cana-1458 35 71 βŠ‚ βŠ‚ PROPN cana-1458 35 72 οΏ½ οΏ½ PROPN cana-1458 35 73 Μƒ Μƒ PROPN cana-1458 35 74 οΏ½ οΏ½ PROPN cana-1458 35 75 β‡’ β‡’ NOUN cana-1458 35 76 β‹ƒΜƒπ‘˜βˆˆπΎ(𝐹 β‹ƒΜƒπ‘˜βˆˆπΎ(𝐹 AUX cana-1458 35 77 , , PUNCT cana-1458 35 78 𝐴)π‘˜ 𝐴)π‘˜ PROPN cana-1458 35 79 ∈ ∈ PROPN cana-1458 35 80 οΏ½ οΏ½ PROPN cana-1458 35 81 Μƒ Μƒ PROPN cana-1458 35 82 οΏ½ οΏ½ PROPN cana-1458 35 83 . . PUNCT cana-1458 36 1 the the DET cana-1458 36 2 pair pair NOUN cana-1458 36 3 ( ( PUNCT cana-1458 36 4 π‘ˆ π‘ˆ PROPN cana-1458 36 5 , , PUNCT cana-1458 36 6 οΏ½ οΏ½ PROPN cana-1458 36 7 Μƒ Μƒ PROPN cana-1458 36 8 οΏ½ οΏ½ PROPN cana-1458 36 9 ) ) PUNCT cana-1458 36 10 is be AUX cana-1458 36 11 called call VERB cana-1458 36 12 a a DET cana-1458 36 13 fuzzy fuzzy ADJ cana-1458 36 14 soft soft ADJ cana-1458 36 15 topological topological ADJ cana-1458 36 16 space space NOUN cana-1458 36 17 . . PUNCT cana-1458 37 1 definition definition NOUN cana-1458 37 2 2.5 2.5 NUM cana-1458 37 3 [ [ SYM cana-1458 37 4 13 13 NUM cana-1458 37 5 ] ] PUNCT cana-1458 37 6 let let VERB cana-1458 37 7 ( ( PUNCT cana-1458 37 8 𝐹 𝐹 PROPN cana-1458 37 9 , , PUNCT cana-1458 37 10 𝐴 𝐴 PROPN cana-1458 37 11 ) ) PUNCT cana-1458 37 12 , , PUNCT cana-1458 37 13 ( ( PUNCT cana-1458 37 14 𝐺 𝐺 NOUN cana-1458 37 15 , , PUNCT cana-1458 37 16 𝐡 𝐡 PROPN cana-1458 37 17 ) ) PUNCT cana-1458 37 18 ∈ ∈ PROPN cana-1458 37 19 𝐹𝑆(π‘ˆ 𝐹𝑆(π‘ˆ NOUN cana-1458 37 20 , , PUNCT cana-1458 37 21 𝐸 𝐸 PROPN cana-1458 37 22 ) ) PUNCT cana-1458 37 23 . . PUNCT cana-1458 38 1 ( ( PUNCT cana-1458 38 2 𝐹 𝐹 PROPN cana-1458 38 3 , , PUNCT cana-1458 38 4 𝐴 𝐴 PROPN cana-1458 38 5 ) ) PUNCT cana-1458 38 6 is be AUX cana-1458 38 7 said say VERB cana-1458 38 8 to to PART cana-1458 38 9 be be AUX cana-1458 38 10 fuzzy fuzzy ADJ cana-1458 38 11 soft soft ADJ cana-1458 38 12 quasi quasi NOUN cana-1458 38 13 - - NOUN cana-1458 38 14 coincident coincident ADJ cana-1458 38 15 with with ADP cana-1458 38 16 ( ( PUNCT cana-1458 38 17 𝐺 𝐺 NOUN cana-1458 38 18 , , PUNCT cana-1458 38 19 𝐡 𝐡 PROPN cana-1458 38 20 ) ) PUNCT cana-1458 38 21 , , PUNCT cana-1458 38 22 denoted denote VERB cana-1458 38 23 by by ADP cana-1458 38 24 ( ( PUNCT cana-1458 38 25 𝐹 𝐹 PROPN cana-1458 38 26 , , PUNCT cana-1458 38 27 𝐴 𝐴 PROPN cana-1458 38 28 ) ) PUNCT cana-1458 38 29 π‘ž(𝐺 π‘ž(𝐺 PROPN cana-1458 38 30 , , PUNCT cana-1458 38 31 𝐡 𝐡 NOUN cana-1458 38 32 ) ) PUNCT cana-1458 38 33 , , PUNCT cana-1458 38 34 if if SCONJ cana-1458 38 35 there there PRON cana-1458 38 36 exists exist VERB cana-1458 38 37 an an DET cana-1458 38 38 𝑒 𝑒 PROPN cana-1458 38 39 ∈ ∈ PROPN cana-1458 38 40 𝐸 𝐸 PROPN cana-1458 38 41 , , PUNCT cana-1458 38 42 π‘₯ π‘₯ DET cana-1458 38 43 ∈ ∈ NOUN cana-1458 38 44 𝑋 𝑋 NOUN cana-1458 38 45 such such ADJ cana-1458 38 46 that that DET cana-1458 38 47 𝐹(𝑒)(π‘₯ 𝐹(𝑒)(π‘₯ NOUN cana-1458 38 48 ) ) PUNCT cana-1458 38 49 + + NUM cana-1458 38 50 𝐺(𝑒)(π‘₯ 𝐺(𝑒)(π‘₯ NOUN cana-1458 38 51 ) ) PUNCT cana-1458 38 52 > > X cana-1458 39 1 1 1 X cana-1458 39 2 . . PUNCT cana-1458 40 1 if if SCONJ cana-1458 40 2 ( ( PUNCT cana-1458 40 3 𝐹 𝐹 PROPN cana-1458 40 4 , , PUNCT cana-1458 40 5 𝐴 𝐴 PROPN cana-1458 40 6 ) ) PUNCT cana-1458 40 7 is be AUX cana-1458 40 8 not not PART cana-1458 40 9 soft soft ADJ cana-1458 40 10 quasi quasi NOUN cana-1458 40 11 - - NOUN cana-1458 40 12 coincident coincident ADJ cana-1458 40 13 with with ADP cana-1458 40 14 ( ( PUNCT cana-1458 40 15 𝐺 𝐺 NOUN cana-1458 40 16 , , PUNCT cana-1458 40 17 𝐡 𝐡 PROPN cana-1458 40 18 ) ) PUNCT cana-1458 40 19 , , PUNCT cana-1458 40 20 then then ADV cana-1458 40 21 we we PRON cana-1458 40 22 write write VERB cana-1458 40 23 ( ( PUNCT cana-1458 40 24 𝐹 𝐹 PROPN cana-1458 40 25 , , PUNCT cana-1458 40 26 𝐴 𝐴 PROPN cana-1458 40 27 ) ) PUNCT cana-1458 40 28 οΏ½ οΏ½ PROPN cana-1458 40 29 Μ… Μ… NOUN cana-1458 40 30 οΏ½ οΏ½ NOUN cana-1458 40 31 (𝐺 (𝐺 SYM cana-1458 40 32 , , PUNCT cana-1458 40 33 𝐡 𝐡 PROPN cana-1458 40 34 ) ) PUNCT cana-1458 40 35 . . PUNCT cana-1458 41 1 definition definition NOUN cana-1458 41 2 2.6 2.6 NUM cana-1458 41 3 [ [ SYM cana-1458 41 4 18 18 NUM cana-1458 41 5 ] ] X cana-1458 41 6 𝐴 𝐴 PROPN cana-1458 41 7 is be AUX cana-1458 41 8 contained contain VERB cana-1458 41 9 in in ADP cana-1458 41 10 𝐡 𝐡 NOUN cana-1458 41 11 ( ( PUNCT cana-1458 41 12 or or CCONJ cana-1458 41 13 , , PUNCT cana-1458 41 14 equivalently equivalently ADV cana-1458 41 15 , , PUNCT cana-1458 41 16 𝐴 𝐴 PROPN cana-1458 41 17 is be AUX cana-1458 41 18 a a DET cana-1458 41 19 subset subset NOUN cana-1458 41 20 of of ADP cana-1458 41 21 𝐡 𝐡 PROPN cana-1458 41 22 , , PUNCT cana-1458 41 23 or or CCONJ cana-1458 41 24 𝐴 𝐴 PROPN cana-1458 41 25 is be AUX cana-1458 41 26 smaller small ADJ cana-1458 41 27 than than ADP cana-1458 41 28 or or CCONJ cana-1458 41 29 equal equal ADJ cana-1458 41 30 to to ADP cana-1458 41 31 𝐡 𝐡 NOUN cana-1458 41 32 ) ) PUNCT cana-1458 41 33 if if SCONJ cana-1458 41 34 and and CCONJ cana-1458 41 35 only only ADV cana-1458 41 36 if if SCONJ cana-1458 41 37 πœ‡π΄ πœ‡π΄ ADP cana-1458 41 38 ≦ ≦ NOUN cana-1458 41 39 πœ‡π΅. πœ‡π΅. NOUN cana-1458 41 40 in in ADP cana-1458 41 41 symbols symbol NOUN cana-1458 41 42 , , PUNCT cana-1458 41 43 π΄βŠ‚π΅ π΄βŠ‚π΅ X cana-1458 41 44 ⇔ ⇔ NOUN cana-1458 41 45 πœ‡π΄ πœ‡π΄ ADP cana-1458 41 46 ≦ ≦ PROPN cana-1458 41 47 πœ‡π΅. πœ‡π΅. NOUN cana-1458 41 48 definition definition NOUN cana-1458 41 49 2.7 2.7 NUM cana-1458 42 1 [ [ X cana-1458 42 2 2 2 NUM cana-1458 42 3 ] ] PUNCT cana-1458 42 4 any any DET cana-1458 42 5 sequence sequence NOUN cana-1458 42 6 of of ADP cana-1458 42 7 subsets subset NOUN cana-1458 42 8 of of ADP cana-1458 42 9 a a DET cana-1458 42 10 non non ADJ cana-1458 42 11 void void NOUN cana-1458 42 12 set set NOUN cana-1458 42 13 𝑋 𝑋 PROPN cana-1458 42 14 is be AUX cana-1458 42 15 called call VERB cana-1458 42 16 a a DET cana-1458 42 17 sequential sequential ADJ cana-1458 42 18 set set NOUN cana-1458 42 19 in in ADP cana-1458 42 20 𝑋. 𝑋. PROPN cana-1458 42 21 that that PRON cana-1458 42 22 is be AUX cana-1458 42 23 , , PUNCT cana-1458 42 24 𝐴(𝑠 𝐴(𝑠 X cana-1458 42 25 ) ) PUNCT cana-1458 42 26 = = SYM cana-1458 42 27 { { PUNCT cana-1458 42 28 𝐴𝑛}𝑛=1 𝐴𝑛}𝑛=1 NOUN cana-1458 42 29 ∞ ∞ PROPN cana-1458 42 30 , , PUNCT cana-1458 42 31 where where SCONJ cana-1458 42 32 each each DET cana-1458 42 33 𝐴𝑛 𝐴𝑛 PROPN cana-1458 42 34 is be AUX cana-1458 42 35 a a DET cana-1458 42 36 subset subset NOUN cana-1458 42 37 of of ADP cana-1458 42 38 𝑋 𝑋 PROPN cana-1458 42 39 , , PUNCT cana-1458 42 40 is be AUX cana-1458 42 41 a a DET cana-1458 42 42 sequential sequential ADJ cana-1458 42 43 set set NOUN cana-1458 42 44 in in ADP cana-1458 42 45 𝑋. 𝑋. PROPN cana-1458 42 46 the the DET cana-1458 42 47 subsets subsets PROPN cana-1458 42 48 𝐴𝑛 𝐴𝑛 PROPN cana-1458 42 49 , , PUNCT cana-1458 42 50 𝑛 𝑛 DET cana-1458 42 51 ∈ ∈ PROPN cana-1458 42 52 β„• β„• PROPN cana-1458 42 53 are be AUX cana-1458 42 54 called call VERB cana-1458 42 55 the the DET cana-1458 42 56 components component NOUN cana-1458 42 57 of of ADP cana-1458 42 58 𝐴(𝑠 𝐴(𝑠 NOUN cana-1458 42 59 ) ) PUNCT cana-1458 42 60 . . PUNCT cana-1458 43 1 definition definition NOUN cana-1458 43 2 2.8[7 2.8[7 NUM cana-1458 43 3 ] ] PUNCT cana-1458 43 4 a a DET cana-1458 43 5 sequence sequence NOUN cana-1458 43 6 of of ADP cana-1458 43 7 fuzzy fuzzy ADJ cana-1458 43 8 soft soft ADJ cana-1458 43 9 sets set NOUN cana-1458 43 10 is be AUX cana-1458 43 11 a a DET cana-1458 43 12 mapping mapping NOUN cana-1458 43 13 from from ADP cana-1458 43 14 β„• β„• PROPN cana-1458 43 15 to to ADP cana-1458 43 16 the the DET cana-1458 43 17 family family NOUN cana-1458 43 18 of of ADP cana-1458 43 19 all all DET cana-1458 43 20 fuzzy fuzzy ADJ cana-1458 43 21 soft soft ADJ cana-1458 43 22 sets set NOUN cana-1458 43 23 and and CCONJ cana-1458 43 24 is be AUX cana-1458 43 25 denoted denote VERB cana-1458 43 26 by by ADP cana-1458 43 27 { { PUNCT cana-1458 43 28 ( ( PUNCT cana-1458 43 29 𝐹 𝐹 PROPN cana-1458 43 30 , , PUNCT cana-1458 43 31 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 43 32 } } PUNCT cana-1458 43 33 or or CCONJ cana-1458 43 34 { { PUNCT cana-1458 43 35 ( ( PUNCT cana-1458 43 36 𝐹 𝐹 PROPN cana-1458 43 37 , , PUNCT cana-1458 43 38 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 43 39 ; ; PUNCT cana-1458 43 40 𝑛 𝑛 PROPN cana-1458 43 41 = = SYM cana-1458 43 42 1,2 1,2 NUM cana-1458 43 43 , , PUNCT cana-1458 43 44 … … PUNCT cana-1458 43 45 } } PUNCT cana-1458 43 46 .that .that PRON cana-1458 43 47 is be AUX cana-1458 43 48 , , PUNCT cana-1458 43 49 { { PUNCT cana-1458 43 50 ( ( PUNCT cana-1458 43 51 𝐹 𝐹 PROPN cana-1458 43 52 , , PUNCT cana-1458 43 53 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 43 54 , , PUNCT cana-1458 43 55 𝑛 𝑛 PROPN cana-1458 43 56 ∈ ∈ PROPN cana-1458 43 57 β„• β„• PROPN cana-1458 43 58 } } PUNCT cana-1458 43 59 where where SCONJ cana-1458 43 60 ( ( PUNCT cana-1458 43 61 𝐹 𝐹 PROPN cana-1458 43 62 , , PUNCT cana-1458 43 63 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 43 64 for for ADP cana-1458 43 65 each each DET cana-1458 43 66 𝑛 𝑛 PRON cana-1458 43 67 ∈ ∈ PROPN cana-1458 43 68 β„• β„• PROPN cana-1458 43 69 represents represent VERB cana-1458 43 70 components component NOUN cana-1458 43 71 of of ADP cana-1458 43 72 fuzzy fuzzy ADJ cana-1458 43 73 soft soft ADJ cana-1458 43 74 set set NOUN cana-1458 43 75 in in ADP cana-1458 43 76 { { PUNCT cana-1458 43 77 ( ( PUNCT cana-1458 43 78 𝐹 𝐹 PROPN cana-1458 43 79 , , PUNCT cana-1458 43 80 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 43 81 } } PUNCT cana-1458 43 82 and and CCONJ cana-1458 43 83 𝑛 𝑛 DET cana-1458 43 84 ∈ ∈ PROPN cana-1458 43 85 β„• β„• PROPN cana-1458 43 86 , , PUNCT cana-1458 43 87 the the DET cana-1458 43 88 set set NOUN cana-1458 43 89 of of ADP cana-1458 43 90 all all DET cana-1458 43 91 natural natural ADJ cana-1458 43 92 numbers number NOUN cana-1458 43 93 . . PUNCT cana-1458 44 1 a a DET cana-1458 44 2 sequence sequence NOUN cana-1458 44 3 of of ADP cana-1458 44 4 fuzzy fuzzy ADJ cana-1458 44 5 soft soft ADJ cana-1458 44 6 sets set NOUN cana-1458 44 7 is be AUX cana-1458 44 8 called call VERB cana-1458 44 9 fuzzy fuzzy ADJ cana-1458 44 10 soft soft ADJ cana-1458 44 11 sequential sequential ADJ cana-1458 44 12 set set NOUN cana-1458 44 13 . . PUNCT cana-1458 45 1 3 3 X cana-1458 45 2 . . X cana-1458 45 3 main main ADJ cana-1458 45 4 results result NOUN cana-1458 45 5 definition definition NOUN cana-1458 45 6 3.1 3.1 NUM cana-1458 45 7 . . PUNCT cana-1458 46 1 a a DET cana-1458 46 2 family family NOUN cana-1458 46 3 οΏ½ οΏ½ PROPN cana-1458 46 4 Μƒ Μƒ PROPN cana-1458 46 5 οΏ½ οΏ½ PROPN cana-1458 46 6 of of ADP cana-1458 46 7 fuzzy fuzzy ADJ cana-1458 46 8 soft soft ADJ cana-1458 46 9 sequential sequential ADJ cana-1458 46 10 sets set NOUN cana-1458 46 11 on on ADP cana-1458 46 12 π‘ˆ π‘ˆ PROPN cana-1458 46 13 satisfying satisfy VERB cana-1458 46 14 the the DET cana-1458 46 15 properties property NOUN cana-1458 46 16 : : PUNCT cana-1458 46 17 communications communication NOUN cana-1458 46 18 on on ADP cana-1458 46 19 applied apply VERB cana-1458 46 20 nonlinear nonlinear ADJ cana-1458 46 21 analysis analysis NOUN cana-1458 46 22 issn issn NOUN cana-1458 46 23 : : PUNCT cana-1458 46 24 1074 1074 NUM cana-1458 46 25 - - PUNCT cana-1458 46 26 133x 133x NUM cana-1458 46 27 vol vol NOUN cana-1458 46 28 31 31 NUM cana-1458 46 29 no no NOUN cana-1458 46 30 . . PUNCT cana-1458 47 1 8s 8s PROPN cana-1458 47 2 ( ( PUNCT cana-1458 47 3 2024 2024 NUM cana-1458 47 4 ) ) PUNCT cana-1458 47 5 89 89 NUM cana-1458 47 6 https://internationalpubls.com https://internationalpubls.com X cana-1458 47 7 ( ( PUNCT cana-1458 47 8 i i NOUN cana-1458 47 9 ) ) PUNCT cana-1458 47 10 βˆ…Μƒ βˆ…Μƒ PROPN cana-1458 47 11 , , PUNCT cana-1458 47 12 οΏ½ οΏ½ PROPN cana-1458 47 13 Μƒ Μƒ PROPN cana-1458 47 14 οΏ½ οΏ½ PROPN cana-1458 47 15 ∈ ∈ PROPN cana-1458 47 16 οΏ½ οΏ½ PROPN cana-1458 47 17 Μƒ Μƒ PROPN cana-1458 47 18 οΏ½ οΏ½ PROPN cana-1458 47 19 ( ( PUNCT cana-1458 47 20 ii ii NOUN cana-1458 47 21 ) ) PUNCT cana-1458 47 22 { { PUNCT cana-1458 47 23 ( ( PUNCT cana-1458 47 24 𝐹 𝐹 PROPN cana-1458 47 25 , , PUNCT cana-1458 47 26 𝐴)𝑛 𝐴)𝑛 PROPN cana-1458 47 27 } } PUNCT cana-1458 47 28 , , PUNCT cana-1458 47 29 { { PUNCT cana-1458 47 30 ( ( PUNCT cana-1458 47 31 𝐺 𝐺 NOUN cana-1458 47 32 , , PUNCT cana-1458 47 33 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 47 34 } } PUNCT cana-1458 47 35 ∈ ∈ PROPN cana-1458 47 36 οΏ½ οΏ½ PROPN cana-1458 47 37 Μƒ Μƒ PROPN cana-1458 47 38 οΏ½ οΏ½ PROPN cana-1458 47 39 β‡’ β‡’ NOUN cana-1458 47 40 { { PUNCT cana-1458 47 41 ( ( PUNCT cana-1458 47 42 𝐹 𝐹 PROPN cana-1458 47 43 , , PUNCT cana-1458 47 44 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 47 45 } } PUNCT cana-1458 47 46 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 47 47 { { PUNCT cana-1458 47 48 ( ( PUNCT cana-1458 47 49 𝐺 𝐺 NOUN cana-1458 47 50 , , PUNCT cana-1458 47 51 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 47 52 } } PUNCT cana-1458 47 53 ∈ ∈ PROPN cana-1458 47 54 οΏ½ οΏ½ PROPN cana-1458 47 55 Μƒ Μƒ PROPN cana-1458 47 56 οΏ½ οΏ½ PROPN cana-1458 47 57 and and CCONJ cana-1458 47 58 ( ( PUNCT cana-1458 47 59 iii iii NOUN cana-1458 47 60 ) ) PUNCT cana-1458 47 61 for for ADP cana-1458 47 62 any any DET cana-1458 47 63 family family NOUN cana-1458 47 64 { { PUNCT cana-1458 47 65 { { PUNCT cana-1458 47 66 ( ( PUNCT cana-1458 47 67 πΉπœ† πΉπœ† NOUN cana-1458 47 68 , , PUNCT cana-1458 47 69 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 47 70 } } PUNCT cana-1458 47 71 , , PUNCT cana-1458 47 72 πœ† πœ† PROPN cana-1458 47 73 ∈ ∈ PROPN cana-1458 47 74 Ξ» Ξ» PROPN cana-1458 47 75 } } PUNCT cana-1458 47 76 ∈ ∈ PROPN cana-1458 47 77 οΏ½ οΏ½ PROPN cana-1458 47 78 Μƒ Μƒ PROPN cana-1458 47 79 οΏ½ οΏ½ PROPN cana-1458 47 80 β‡’ β‡’ NOUN cana-1458 47 81 { { PUNCT cana-1458 47 82 ( ( PUNCT cana-1458 47 83 πΉπœ† πΉπœ† NOUN cana-1458 47 84 , , PUNCT cana-1458 47 85 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 47 86 } } PUNCT cana-1458 47 87 ∈ ∈ PROPN cana-1458 47 88 οΏ½ οΏ½ PROPN cana-1458 47 89 Μƒ Μƒ PROPN cana-1458 47 90 οΏ½ οΏ½ PROPN cana-1458 47 91 πœ†βˆˆΞ» πœ†βˆˆΞ» PROPN cana-1458 47 92 βˆͺΜƒ βˆͺΜƒ PROPN cana-1458 47 93 is be AUX cana-1458 47 94 called call VERB cana-1458 47 95 a a DET cana-1458 47 96 fuzzy fuzzy ADJ cana-1458 47 97 soft soft ADJ cana-1458 47 98 sequential sequential ADJ cana-1458 47 99 topology topology NOUN cana-1458 47 100 on on ADP cana-1458 47 101 π‘ˆ π‘ˆ PROPN cana-1458 47 102 and and CCONJ cana-1458 47 103 the the DET cana-1458 47 104 triplet triplet NOUN cana-1458 47 105 ( ( PUNCT cana-1458 47 106 π‘ˆ π‘ˆ PROPN cana-1458 47 107 , , PUNCT cana-1458 47 108 οΏ½ οΏ½ PROPN cana-1458 47 109 Μƒ Μƒ PROPN cana-1458 47 110 οΏ½ οΏ½ PROPN cana-1458 47 111 , , PUNCT cana-1458 47 112 𝐸 𝐸 PROPN cana-1458 47 113 ) ) PUNCT cana-1458 47 114 is be AUX cana-1458 47 115 called call VERB cana-1458 47 116 a a DET cana-1458 47 117 fuzzy fuzzy ADJ cana-1458 47 118 soft soft ADJ cana-1458 47 119 sequential sequential ADJ cana-1458 47 120 topological topological ADJ cana-1458 47 121 space space NOUN cana-1458 47 122 over over ADP cana-1458 47 123 π‘ˆ π‘ˆ PROPN cana-1458 47 124 ( ( PUNCT cana-1458 47 125 briefly briefly ADV cana-1458 47 126 , , PUNCT cana-1458 47 127 𝐹𝑆𝑆𝑇𝑆 𝐹𝑆𝑆𝑇𝑆 PROPN cana-1458 47 128 ) ) PUNCT cana-1458 47 129 . . PUNCT cana-1458 48 1 definition definition NOUN cana-1458 48 2 3.2 3.2 NUM cana-1458 48 3 . . PUNCT cana-1458 49 1 let let AUX cana-1458 49 2 ( ( PUNCT cana-1458 49 3 π‘ˆ π‘ˆ PROPN cana-1458 49 4 , , PUNCT cana-1458 49 5 οΏ½ οΏ½ PROPN cana-1458 49 6 Μƒ Μƒ PROPN cana-1458 49 7 οΏ½ οΏ½ PROPN cana-1458 49 8 , , PUNCT cana-1458 49 9 𝐸 𝐸 PROPN cana-1458 49 10 ) ) PUNCT cana-1458 49 11 be be VERB cana-1458 49 12 a a DET cana-1458 49 13 fuzzy fuzzy ADJ cana-1458 49 14 soft soft ADJ cana-1458 49 15 sequential sequential ADJ cana-1458 49 16 topology topology NOUN cana-1458 49 17 on on ADP cana-1458 49 18 π‘ˆ. π‘ˆ. PROPN cana-1458 49 19 then then ADV cana-1458 49 20 the the DET cana-1458 49 21 members member NOUN cana-1458 49 22 of of ADP cana-1458 49 23 οΏ½ οΏ½ PROPN cana-1458 49 24 Μƒ Μƒ PROPN cana-1458 49 25 οΏ½ οΏ½ PROPN cana-1458 49 26 are be AUX cana-1458 49 27 called call VERB cana-1458 49 28 fuzzy fuzzy ADJ cana-1458 49 29 soft soft ADJ cana-1458 49 30 sequential sequential ADJ cana-1458 49 31 open open ADJ cana-1458 49 32 sets set NOUN cana-1458 49 33 ( ( PUNCT cana-1458 49 34 𝐹𝑆𝑆-open 𝐹𝑆𝑆-open ADJ cana-1458 49 35 sets set NOUN cana-1458 49 36 ) ) PUNCT cana-1458 49 37 . . PUNCT cana-1458 50 1 the the DET cana-1458 50 2 complement complement NOUN cana-1458 50 3 of of ADP cana-1458 50 4 a a DET cana-1458 50 5 fuzzy fuzzy ADJ cana-1458 50 6 soft soft ADJ cana-1458 50 7 sequential sequential ADJ cana-1458 50 8 open open ADJ cana-1458 50 9 set set NOUN cana-1458 50 10 is be AUX cana-1458 50 11 called call VERB cana-1458 50 12 fuzzy fuzzy ADJ cana-1458 50 13 soft soft ADJ cana-1458 50 14 sequential sequential ADJ cana-1458 50 15 closed close VERB cana-1458 50 16 set set NOUN cana-1458 50 17 ( ( PUNCT cana-1458 50 18 𝐹𝑆𝑆 𝐹𝑆𝑆 PROPN cana-1458 50 19 -closed -close VERB cana-1458 50 20 set set NOUN cana-1458 50 21 ) ) PUNCT cana-1458 50 22 . . PUNCT cana-1458 50 23 example example NOUN cana-1458 51 1 3.3 3.3 NUM cana-1458 51 2 . . PUNCT cana-1458 52 1 let let VERB cana-1458 52 2 us we PRON cana-1458 52 3 consider consider VERB cana-1458 52 4 the the DET cana-1458 52 5 universe universe NOUN cana-1458 52 6 π‘ˆ π‘ˆ PROPN cana-1458 52 7 = = SYM cana-1458 52 8 { { PUNCT cana-1458 52 9 π‘₯1 π‘₯1 NOUN cana-1458 52 10 , , PUNCT cana-1458 52 11 π‘₯2 π‘₯2 NOUN cana-1458 52 12 } } PUNCT cana-1458 52 13 and and CCONJ cana-1458 52 14 𝐴 𝐴 PROPN cana-1458 52 15 = = SYM cana-1458 52 16 { { PUNCT cana-1458 52 17 𝑒1 𝑒1 NOUN cana-1458 52 18 , , PUNCT cana-1458 52 19 𝑒2 𝑒2 NOUN cana-1458 52 20 } } PUNCT cana-1458 52 21 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 52 22 𝐸 𝐸 PROPN cana-1458 52 23 where where SCONJ cana-1458 52 24 { { PUNCT cana-1458 52 25 𝑒1 𝑒1 NOUN cana-1458 52 26 , , PUNCT cana-1458 52 27 𝑒2 𝑒2 PROPN cana-1458 52 28 } } PUNCT cana-1458 52 29 be be AUX cana-1458 52 30 a a DET cana-1458 52 31 collection collection NOUN cana-1458 52 32 of of ADP cana-1458 52 33 sets set NOUN cana-1458 52 34 of of ADP cana-1458 52 35 parameters parameter NOUN cana-1458 52 36 . . PUNCT cana-1458 53 1 let let VERB cana-1458 53 2 us we PRON cana-1458 53 3 take take VERB cana-1458 53 4 a a DET cana-1458 53 5 fuzzy fuzzy ADJ cana-1458 53 6 soft soft ADJ cana-1458 53 7 sequential sequential ADJ cana-1458 53 8 topological topological ADJ cana-1458 53 9 space space NOUN cana-1458 53 10 ( ( PUNCT cana-1458 53 11 π‘ˆ π‘ˆ PROPN cana-1458 53 12 , , PUNCT cana-1458 53 13 οΏ½ οΏ½ PROPN cana-1458 53 14 Μƒ Μƒ PROPN cana-1458 53 15 οΏ½ οΏ½ PROPN cana-1458 53 16 , , PUNCT cana-1458 53 17 𝐸 𝐸 PROPN cana-1458 53 18 ) ) PUNCT cana-1458 54 1 where where SCONJ cana-1458 54 2 οΏ½ οΏ½ PROPN cana-1458 54 3 Μƒ Μƒ PROPN cana-1458 54 4 οΏ½ οΏ½ NOUN cana-1458 54 5 = = SYM cana-1458 54 6 { { PUNCT cana-1458 54 7 βˆ…Μƒ βˆ…Μƒ NOUN cana-1458 54 8 , , PUNCT cana-1458 54 9 οΏ½ οΏ½ PROPN cana-1458 54 10 Μƒ Μƒ PROPN cana-1458 54 11 οΏ½ οΏ½ PROPN cana-1458 54 12 , , PUNCT cana-1458 54 13 { { PUNCT cana-1458 54 14 ( ( PUNCT cana-1458 54 15 𝐹 𝐹 PROPN cana-1458 54 16 , , PUNCT cana-1458 54 17 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 54 18 } } PUNCT cana-1458 54 19 } } PUNCT cana-1458 54 20 . . PUNCT cana-1458 55 1 the the DET cana-1458 55 2 nth nth PROPN cana-1458 55 3 component component NOUN cana-1458 55 4 of of ADP cana-1458 55 5 the the DET cana-1458 55 6 fuzzy fuzzy ADJ cana-1458 55 7 soft soft ADJ cana-1458 55 8 sequential sequential ADJ cana-1458 55 9 set set NOUN cana-1458 55 10 { { PUNCT cana-1458 55 11 ( ( PUNCT cana-1458 55 12 𝐹 𝐹 PROPN cana-1458 55 13 , , PUNCT cana-1458 55 14 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 55 15 } } PUNCT cana-1458 55 16 is be AUX cana-1458 55 17 defined define VERB cana-1458 55 18 as as ADP cana-1458 55 19 ( ( PUNCT cana-1458 55 20 𝐹 𝐹 PROPN cana-1458 55 21 , , PUNCT cana-1458 55 22 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 56 1 = = PRON cana-1458 56 2 { { PUNCT cana-1458 56 3 ( ( PUNCT cana-1458 56 4 𝑒1 𝑒1 NOUN cana-1458 56 5 , , PUNCT cana-1458 56 6 { { PUNCT cana-1458 56 7 π‘₯1 π‘₯1 NOUN cana-1458 56 8 ( ( PUNCT cana-1458 56 9 1 1 NUM cana-1458 56 10 3𝑛)⁄ 3𝑛)⁄ NUM cana-1458 56 11 , , PUNCT cana-1458 56 12 π‘₯2 π‘₯2 NOUN cana-1458 56 13 ( ( PUNCT cana-1458 56 14 𝑛 𝑛 NOUN cana-1458 56 15 𝑛+2)⁄ 𝑛+2)⁄ NOUN cana-1458 56 16 } } PUNCT cana-1458 56 17 ) ) PUNCT cana-1458 56 18 , , PUNCT cana-1458 56 19 ( ( PUNCT cana-1458 56 20 𝑒2 𝑒2 NOUN cana-1458 56 21 , , PUNCT cana-1458 56 22 { { PUNCT cana-1458 56 23 π‘₯1 π‘₯1 NOUN cana-1458 56 24 ( ( PUNCT cana-1458 56 25 2 2 NUM cana-1458 56 26 5𝑛)⁄ 5𝑛)⁄ NUM cana-1458 56 27 , , PUNCT cana-1458 56 28 π‘₯2 π‘₯2 NOUN cana-1458 56 29 ( ( PUNCT cana-1458 56 30 1 1 NUM cana-1458 56 31 𝑛)⁄ 𝑛)⁄ NOUN cana-1458 56 32 } } PUNCT cana-1458 56 33 ) ) PUNCT cana-1458 56 34 } } PUNCT cana-1458 56 35 . . PUNCT cana-1458 57 1 then then ADV cana-1458 57 2 οΏ½ οΏ½ PROPN cana-1458 57 3 Μƒ Μƒ PROPN cana-1458 57 4 οΏ½ οΏ½ PROPN cana-1458 57 5 is be AUX cana-1458 57 6 a a DET cana-1458 57 7 fuzzy fuzzy ADJ cana-1458 57 8 soft soft ADJ cana-1458 57 9 sequential sequential ADJ cana-1458 57 10 topology topology NOUN cana-1458 57 11 on on ADP cana-1458 57 12 π‘ˆ. π‘ˆ. PROPN cana-1458 57 13 definition definition NOUN cana-1458 57 14 3.4 3.4 NUM cana-1458 57 15 . . PUNCT cana-1458 58 1 let let VERB cana-1458 58 2 π‘ˆ π‘ˆ PROPN cana-1458 58 3 be be AUX cana-1458 58 4 the the DET cana-1458 58 5 universal universal ADJ cana-1458 58 6 set set NOUN cana-1458 58 7 and and CCONJ cana-1458 58 8 𝐸 𝐸 PROPN cana-1458 58 9 be be VERB cana-1458 58 10 a a DET cana-1458 58 11 set set NOUN cana-1458 58 12 of of ADP cana-1458 58 13 parameters parameter NOUN cana-1458 58 14 and and CCONJ cana-1458 58 15 οΏ½ οΏ½ PROPN cana-1458 58 16 Μƒ Μƒ PROPN cana-1458 58 17 οΏ½ οΏ½ NOUN cana-1458 58 18 = = SYM cana-1458 58 19 { { PUNCT cana-1458 58 20 βˆ…Μƒ βˆ…Μƒ NOUN cana-1458 58 21 , , PUNCT cana-1458 58 22 οΏ½ οΏ½ PROPN cana-1458 58 23 Μƒ Μƒ PROPN cana-1458 58 24 οΏ½ οΏ½ NOUN cana-1458 58 25 } } PUNCT cana-1458 58 26 . . PUNCT cana-1458 59 1 then then ADV cana-1458 59 2 οΏ½ οΏ½ PROPN cana-1458 59 3 Μƒ Μƒ PROPN cana-1458 59 4 οΏ½ οΏ½ PROPN cana-1458 59 5 is be AUX cana-1458 59 6 called call VERB cana-1458 59 7 the the DET cana-1458 59 8 fuzzy fuzzy ADJ cana-1458 59 9 soft soft ADJ cana-1458 59 10 sequential sequential ADJ cana-1458 59 11 indiscrete indiscrete ADJ cana-1458 59 12 topology topology NOUN cana-1458 59 13 on on ADP cana-1458 59 14 π‘ˆ. π‘ˆ. PROPN cana-1458 59 15 definition definition NOUN cana-1458 59 16 3.5 3.5 NUM cana-1458 59 17 . . PUNCT cana-1458 60 1 let let VERB cana-1458 60 2 π‘ˆ π‘ˆ PROPN cana-1458 60 3 be be AUX cana-1458 60 4 the the DET cana-1458 60 5 universal universal ADJ cana-1458 60 6 set set NOUN cana-1458 60 7 and and CCONJ cana-1458 60 8 e e NOUN cana-1458 60 9 be be AUX cana-1458 60 10 a a DET cana-1458 60 11 set set NOUN cana-1458 60 12 of of ADP cana-1458 60 13 parameters parameter NOUN cana-1458 60 14 and and CCONJ cana-1458 60 15 let let VERB cana-1458 60 16 οΏ½ οΏ½ PROPN cana-1458 60 17 Μƒ Μƒ PROPN cana-1458 60 18 οΏ½ οΏ½ PROPN cana-1458 60 19 be be AUX cana-1458 60 20 the the DET cana-1458 60 21 collection collection NOUN cana-1458 60 22 of of ADP cana-1458 60 23 all all DET cana-1458 60 24 fuzzy fuzzy ADJ cana-1458 60 25 soft soft ADJ cana-1458 60 26 sequential sequential ADJ cana-1458 60 27 sets set NOUN cana-1458 60 28 which which PRON cana-1458 60 29 can can AUX cana-1458 60 30 be be AUX cana-1458 60 31 defined define VERB cana-1458 60 32 over over ADP cana-1458 60 33 π‘ˆ. π‘ˆ. PROPN cana-1458 60 34 then then ADV cana-1458 60 35 οΏ½ οΏ½ PROPN cana-1458 60 36 Μƒ Μƒ PROPN cana-1458 60 37 οΏ½ οΏ½ PROPN cana-1458 60 38 is be AUX cana-1458 60 39 called call VERB cana-1458 60 40 the the DET cana-1458 60 41 fuzzy fuzzy ADJ cana-1458 60 42 soft soft ADJ cana-1458 60 43 sequential sequential ADJ cana-1458 60 44 discrete discrete ADJ cana-1458 60 45 topology topology NOUN cana-1458 60 46 on on ADP cana-1458 60 47 π‘ˆ. π‘ˆ. PROPN cana-1458 60 48 definition definition NOUN cana-1458 60 49 3.6 3.6 NUM cana-1458 60 50 . . PUNCT cana-1458 61 1 { { PUNCT cana-1458 61 2 ( ( PUNCT cana-1458 61 3 𝐹 𝐹 PROPN cana-1458 61 4 , , PUNCT cana-1458 61 5 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 61 6 } } PUNCT cana-1458 61 7 is be AUX cana-1458 61 8 contained contain VERB cana-1458 61 9 in in ADP cana-1458 61 10 { { PUNCT cana-1458 61 11 ( ( PUNCT cana-1458 61 12 𝐺 𝐺 NOUN cana-1458 61 13 , , PUNCT cana-1458 61 14 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 61 15 } } PUNCT cana-1458 61 16 , , PUNCT cana-1458 61 17 symbolically symbolically ADV cana-1458 61 18 { { PUNCT cana-1458 61 19 ( ( PUNCT cana-1458 61 20 𝐹 𝐹 PROPN cana-1458 61 21 , , PUNCT cana-1458 61 22 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 61 23 } } PUNCT cana-1458 61 24 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 61 25 { { PUNCT cana-1458 61 26 ( ( PUNCT cana-1458 61 27 𝐺 𝐺 NOUN cana-1458 61 28 , , PUNCT cana-1458 61 29 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 61 30 } } PUNCT cana-1458 61 31 , , PUNCT cana-1458 61 32 if if SCONJ cana-1458 61 33 and and CCONJ cana-1458 61 34 only only ADV cana-1458 61 35 if if SCONJ cana-1458 61 36 for for ADP cana-1458 61 37 each each DET cana-1458 61 38 𝑒𝑖 𝑒𝑖 PRON cana-1458 61 39 ∈ ∈ PROPN cana-1458 61 40 𝐴 𝐴 PROPN cana-1458 61 41 , , PUNCT cana-1458 61 42 𝐡 𝐡 NOUN cana-1458 61 43 and and CCONJ cana-1458 61 44 for for ADP cana-1458 61 45 all all DET cana-1458 61 46 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 61 47 ∈ ∈ PROPN cana-1458 61 48 π‘ˆ π‘ˆ PROPN cana-1458 61 49 , , PUNCT cana-1458 61 50 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 61 51 ) ) PUNCT cana-1458 61 52 ≀̃ ≀̃ NOUN cana-1458 61 53 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 61 54 ) ) PUNCT cana-1458 61 55 for for ADP cana-1458 61 56 all all DET cana-1458 61 57 𝑛 𝑛 DET cana-1458 61 58 ∈ ∈ PROPN cana-1458 61 59 β„• β„• PROPN cana-1458 61 60 . . PUNCT cana-1458 62 1 definition definition NOUN cana-1458 62 2 3.7 3.7 NUM cana-1458 62 3 . . PUNCT cana-1458 63 1 { { PUNCT cana-1458 63 2 ( ( PUNCT cana-1458 63 3 𝐹 𝐹 PROPN cana-1458 63 4 , , PUNCT cana-1458 63 5 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 63 6 } } PUNCT cana-1458 63 7 is be AUX cana-1458 63 8 weakly weakly ADV cana-1458 63 9 contained contain VERB cana-1458 63 10 in in ADP cana-1458 63 11 { { PUNCT cana-1458 63 12 ( ( PUNCT cana-1458 63 13 𝐺 𝐺 NOUN cana-1458 63 14 , , PUNCT cana-1458 63 15 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 63 16 } } PUNCT cana-1458 63 17 , , PUNCT cana-1458 63 18 symbolically symbolically ADV cana-1458 63 19 { { PUNCT cana-1458 63 20 ( ( PUNCT cana-1458 63 21 𝐹 𝐹 PROPN cana-1458 63 22 , , PUNCT cana-1458 63 23 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 63 24 } } PUNCT cana-1458 63 25 βŠ‚Μƒπ‘€ βŠ‚Μƒπ‘€ NOUN cana-1458 63 26 { { PUNCT cana-1458 63 27 ( ( PUNCT cana-1458 63 28 𝐺 𝐺 NOUN cana-1458 63 29 , , PUNCT cana-1458 63 30 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 63 31 } } PUNCT cana-1458 63 32 , , PUNCT cana-1458 63 33 if if SCONJ cana-1458 63 34 and and CCONJ cana-1458 63 35 only only ADV cana-1458 63 36 if if SCONJ cana-1458 63 37 for for ADP cana-1458 63 38 each each DET cana-1458 63 39 𝑒𝑖 𝑒𝑖 PRON cana-1458 63 40 ∈ ∈ PROPN cana-1458 63 41 𝐴 𝐴 PROPN cana-1458 63 42 , , PUNCT cana-1458 63 43 𝐡 𝐡 NOUN cana-1458 63 44 and and CCONJ cana-1458 63 45 for for ADP cana-1458 63 46 all all DET cana-1458 63 47 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 63 48 ∈ ∈ PROPN cana-1458 63 49 π‘ˆ π‘ˆ PROPN cana-1458 63 50 , , PUNCT cana-1458 63 51 there there PRON cana-1458 63 52 exists exist VERB cana-1458 63 53 𝑛 𝑛 DET cana-1458 63 54 ∈ ∈ PROPN cana-1458 63 55 β„• β„• PROPN cana-1458 63 56 such such ADJ cana-1458 63 57 that that SCONJ cana-1458 63 58 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 63 59 ) ) PUNCT cana-1458 63 60 ≀̃ ≀̃ NOUN cana-1458 63 61 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 63 62 ) ) PUNCT cana-1458 63 63 . . PUNCT cana-1458 64 1 definition definition NOUN cana-1458 64 2 3.8 3.8 NUM cana-1458 64 3 . . PUNCT cana-1458 65 1 let let VERB cana-1458 65 2 ( ( PUNCT cana-1458 65 3 π‘ˆ π‘ˆ PROPN cana-1458 65 4 , , PUNCT cana-1458 65 5 οΏ½ οΏ½ PROPN cana-1458 65 6 Μƒ Μƒ NOUN cana-1458 65 7 οΏ½ οΏ½ NOUN cana-1458 65 8 1 1 NUM cana-1458 65 9 , , PUNCT cana-1458 65 10 𝐸 𝐸 PROPN cana-1458 65 11 ) ) PUNCT cana-1458 65 12 and and CCONJ cana-1458 65 13 ( ( PUNCT cana-1458 65 14 π‘ˆ π‘ˆ PROPN cana-1458 65 15 , , PUNCT cana-1458 65 16 οΏ½ οΏ½ PROPN cana-1458 65 17 Μƒ Μƒ NOUN cana-1458 65 18 οΏ½ οΏ½ NOUN cana-1458 65 19 2 2 NUM cana-1458 65 20 , , PUNCT cana-1458 65 21 𝐸 𝐸 PROPN cana-1458 65 22 ) ) PUNCT cana-1458 65 23 be be VERB cana-1458 65 24 two two NUM cana-1458 65 25 fuzzy fuzzy ADJ cana-1458 65 26 soft soft ADJ cana-1458 65 27 sequential sequential ADJ cana-1458 65 28 topological topological ADJ cana-1458 65 29 spaces space NOUN cana-1458 65 30 . . PUNCT cana-1458 66 1 then then ADV cana-1458 66 2 the the DET cana-1458 66 3 following follow VERB cana-1458 66 4 holds hold VERB cana-1458 66 5 : : PUNCT cana-1458 66 6 ( ( PUNCT cana-1458 66 7 i i NOUN cana-1458 66 8 ) ) PUNCT cana-1458 66 9 if if SCONJ cana-1458 66 10 οΏ½ οΏ½ PROPN cana-1458 66 11 Μƒ Μƒ PROPN cana-1458 66 12 οΏ½ οΏ½ SYM cana-1458 66 13 1 1 NUM cana-1458 66 14 βŠ‚Μƒ βŠ‚Μƒ PROPN cana-1458 66 15 οΏ½ οΏ½ PROPN cana-1458 66 16 Μƒ Μƒ NOUN cana-1458 66 17 οΏ½ οΏ½ NOUN cana-1458 66 18 2 2 NUM cana-1458 66 19 , , PUNCT cana-1458 66 20 then then ADV cana-1458 66 21 οΏ½ οΏ½ PROPN cana-1458 66 22 Μƒ Μƒ PROPN cana-1458 66 23 οΏ½ οΏ½ NOUN cana-1458 66 24 2 2 NUM cana-1458 66 25 is be AUX cana-1458 66 26 fuzzy fuzzy ADJ cana-1458 66 27 soft soft ADJ cana-1458 66 28 sequential sequential ADJ cana-1458 66 29 finer fine ADJ cana-1458 66 30 than than ADP cana-1458 66 31 οΏ½ οΏ½ PROPN cana-1458 66 32 Μƒ Μƒ PROPN cana-1458 66 33 οΏ½ οΏ½ PROPN cana-1458 66 34 1 1 NUM cana-1458 66 35 or or CCONJ cana-1458 66 36 οΏ½ οΏ½ PROPN cana-1458 66 37 Μƒ Μƒ PROPN cana-1458 66 38 οΏ½ οΏ½ NOUN cana-1458 66 39 1 1 NUM cana-1458 66 40 is be AUX cana-1458 66 41 fuzzy fuzzy ADJ cana-1458 66 42 soft soft ADJ cana-1458 66 43 sequential sequential ADJ cana-1458 66 44 coarser coarse ADJ cana-1458 66 45 than than ADP cana-1458 66 46 οΏ½ οΏ½ PROPN cana-1458 66 47 Μƒ Μƒ NOUN cana-1458 66 48 οΏ½ οΏ½ PROPN cana-1458 66 49 2 2 NUM cana-1458 66 50 . . PUNCT cana-1458 66 51 ( ( PUNCT cana-1458 66 52 ii ii NOUN cana-1458 66 53 ) ) PUNCT cana-1458 66 54 if if SCONJ cana-1458 66 55 either either DET cana-1458 66 56 οΏ½ οΏ½ PROPN cana-1458 66 57 Μƒ Μƒ PROPN cana-1458 66 58 οΏ½ οΏ½ NOUN cana-1458 66 59 1 1 NUM cana-1458 66 60 βŠ‚Μƒ βŠ‚Μƒ PROPN cana-1458 66 61 οΏ½ οΏ½ PROPN cana-1458 66 62 Μƒ Μƒ NOUN cana-1458 66 63 οΏ½ οΏ½ PROPN cana-1458 66 64 2 2 NUM cana-1458 66 65 or or CCONJ cana-1458 66 66 οΏ½ οΏ½ PROPN cana-1458 66 67 Μƒ Μƒ PROPN cana-1458 66 68 οΏ½ οΏ½ NOUN cana-1458 66 69 2 2 NUM cana-1458 66 70 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 66 71 οΏ½ οΏ½ PROPN cana-1458 66 72 Μƒ Μƒ NOUN cana-1458 66 73 οΏ½ οΏ½ NOUN cana-1458 66 74 1 1 NUM cana-1458 66 75 , , PUNCT cana-1458 66 76 then then ADV cana-1458 66 77 οΏ½ οΏ½ PROPN cana-1458 66 78 Μƒ Μƒ PROPN cana-1458 66 79 οΏ½ οΏ½ NOUN cana-1458 66 80 1 1 NUM cana-1458 66 81 is be AUX cana-1458 66 82 comparable comparable ADJ cana-1458 66 83 with with ADP cana-1458 66 84 οΏ½ οΏ½ PROPN cana-1458 66 85 Μƒ Μƒ NOUN cana-1458 66 86 οΏ½ οΏ½ NOUN cana-1458 66 87 2 2 NUM cana-1458 66 88 . . PUNCT cana-1458 67 1 theorem theorem VERB cana-1458 67 2 3.9 3.9 NUM cana-1458 67 3 let let VERB cana-1458 67 4 ( ( PUNCT cana-1458 67 5 π‘ˆ π‘ˆ PROPN cana-1458 67 6 , , PUNCT cana-1458 67 7 οΏ½ οΏ½ PROPN cana-1458 67 8 Μƒ Μƒ NOUN cana-1458 67 9 οΏ½ οΏ½ NOUN cana-1458 67 10 1 1 NUM cana-1458 67 11 , , PUNCT cana-1458 67 12 𝐸 𝐸 PROPN cana-1458 67 13 ) ) PUNCT cana-1458 67 14 and and CCONJ cana-1458 67 15 ( ( PUNCT cana-1458 67 16 π‘ˆ π‘ˆ PROPN cana-1458 67 17 , , PUNCT cana-1458 67 18 οΏ½ οΏ½ PROPN cana-1458 67 19 Μƒ Μƒ NOUN cana-1458 67 20 οΏ½ οΏ½ NOUN cana-1458 67 21 2 2 NUM cana-1458 67 22 , , PUNCT cana-1458 67 23 𝐸 𝐸 PROPN cana-1458 67 24 ) ) PUNCT cana-1458 67 25 be be VERB cana-1458 67 26 two two NUM cana-1458 67 27 fuzzy fuzzy ADJ cana-1458 67 28 soft soft ADJ cana-1458 67 29 sequential sequential ADJ cana-1458 67 30 topological topological ADJ cana-1458 67 31 spaces space NOUN cana-1458 67 32 over over ADP cana-1458 67 33 π‘ˆ π‘ˆ PROPN cana-1458 67 34 , , PUNCT cana-1458 67 35 then then ADV cana-1458 67 36 ( ( PUNCT cana-1458 67 37 π‘ˆ π‘ˆ PROPN cana-1458 67 38 , , PUNCT cana-1458 67 39 οΏ½ οΏ½ PROPN cana-1458 67 40 Μƒ Μƒ NOUN cana-1458 67 41 οΏ½ οΏ½ PROPN cana-1458 67 42 1 1 NUM cana-1458 67 43 βˆ©Μƒ βˆ©Μƒ SYM cana-1458 67 44 οΏ½ οΏ½ PROPN cana-1458 67 45 Μƒ Μƒ PROPN cana-1458 67 46 οΏ½ οΏ½ NOUN cana-1458 67 47 2 2 NUM cana-1458 67 48 , , PUNCT cana-1458 67 49 𝐸 𝐸 PROPN cana-1458 67 50 ) ) PUNCT cana-1458 67 51 is be AUX cana-1458 67 52 a a DET cana-1458 67 53 fuzzy fuzzy ADJ cana-1458 67 54 soft soft ADJ cana-1458 67 55 sequential sequential ADJ cana-1458 67 56 topological topological ADJ cana-1458 67 57 space space NOUN cana-1458 67 58 overπ‘ˆ. overπ‘ˆ. NOUN cana-1458 67 59 proof proof NOUN cana-1458 67 60 . . PUNCT cana-1458 68 1 ( ( PUNCT cana-1458 68 2 i i NOUN cana-1458 68 3 ) ) PUNCT cana-1458 68 4 βˆ…Μƒ βˆ…Μƒ PROPN cana-1458 68 5 , , PUNCT cana-1458 68 6 οΏ½ οΏ½ PROPN cana-1458 68 7 Μƒ Μƒ PROPN cana-1458 68 8 οΏ½ οΏ½ PROPN cana-1458 68 9 ∈ ∈ PROPN cana-1458 68 10 οΏ½ οΏ½ PROPN cana-1458 68 11 Μƒ Μƒ PROPN cana-1458 68 12 οΏ½ οΏ½ PROPN cana-1458 68 13 1 1 NUM cana-1458 68 14 βˆ©Μƒ βˆ©Μƒ SYM cana-1458 68 15 οΏ½ οΏ½ PROPN cana-1458 68 16 Μƒ Μƒ PROPN cana-1458 68 17 οΏ½ οΏ½ PROPN cana-1458 68 18 2 2 NUM cana-1458 68 19 . . PUNCT cana-1458 68 20 ( ( PUNCT cana-1458 68 21 ii ii NOUN cana-1458 68 22 ) ) PUNCT cana-1458 68 23 let let VERB cana-1458 68 24 the the DET cana-1458 68 25 two two NUM cana-1458 68 26 fuzzy fuzzy ADJ cana-1458 68 27 soft soft ADJ cana-1458 68 28 sequential sequential ADJ cana-1458 68 29 sets set NOUN cana-1458 68 30 { { PUNCT cana-1458 68 31 ( ( PUNCT cana-1458 68 32 𝐹 𝐹 PROPN cana-1458 68 33 , , PUNCT cana-1458 68 34 𝐴)𝑛 𝐴)𝑛 PROPN cana-1458 68 35 } } PUNCT cana-1458 68 36 , , PUNCT cana-1458 68 37 { { PUNCT cana-1458 68 38 ( ( PUNCT cana-1458 68 39 𝐺 𝐺 NOUN cana-1458 68 40 , , PUNCT cana-1458 68 41 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 68 42 } } PUNCT cana-1458 68 43 ∈ ∈ PROPN cana-1458 68 44 οΏ½ οΏ½ PROPN cana-1458 68 45 Μƒ Μƒ PROPN cana-1458 68 46 οΏ½ οΏ½ PROPN cana-1458 68 47 1 1 NUM cana-1458 68 48 βˆ©Μƒ βˆ©Μƒ SYM cana-1458 68 49 οΏ½ οΏ½ PROPN cana-1458 68 50 Μƒ Μƒ PROPN cana-1458 68 51 οΏ½ οΏ½ NOUN cana-1458 68 52 2 2 NUM cana-1458 68 53 . . PUNCT cana-1458 69 1 then then ADV cana-1458 69 2 { { PUNCT cana-1458 69 3 ( ( PUNCT cana-1458 69 4 𝐹 𝐹 PROPN cana-1458 69 5 , , PUNCT cana-1458 69 6 𝐴)𝑛 𝐴)𝑛 PROPN cana-1458 69 7 } } PUNCT cana-1458 69 8 , , PUNCT cana-1458 69 9 { { PUNCT cana-1458 69 10 ( ( PUNCT cana-1458 69 11 𝐺 𝐺 NOUN cana-1458 69 12 , , PUNCT cana-1458 69 13 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 69 14 } } PUNCT cana-1458 69 15 ∈ ∈ PROPN cana-1458 69 16 οΏ½ οΏ½ PROPN cana-1458 69 17 Μƒ Μƒ PROPN cana-1458 69 18 οΏ½ οΏ½ NOUN cana-1458 69 19 1 1 NUM cana-1458 69 20 and and CCONJ cana-1458 69 21 { { PUNCT cana-1458 69 22 ( ( PUNCT cana-1458 69 23 𝐹 𝐹 PROPN cana-1458 69 24 , , PUNCT cana-1458 69 25 𝐴)𝑛 𝐴)𝑛 PROPN cana-1458 69 26 } } PUNCT cana-1458 69 27 , , PUNCT cana-1458 69 28 { { PUNCT cana-1458 69 29 ( ( PUNCT cana-1458 69 30 𝐺 𝐺 NOUN cana-1458 69 31 , , PUNCT cana-1458 69 32 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 69 33 } } PUNCT cana-1458 69 34 ∈ ∈ PROPN cana-1458 69 35 οΏ½ οΏ½ PROPN cana-1458 69 36 Μƒ Μƒ PROPN cana-1458 69 37 οΏ½ οΏ½ NOUN cana-1458 69 38 2 2 NUM cana-1458 69 39 . . PUNCT cana-1458 70 1 therefore therefore ADV cana-1458 70 2 { { PUNCT cana-1458 70 3 ( ( PUNCT cana-1458 70 4 𝐹 𝐹 PROPN cana-1458 70 5 , , PUNCT cana-1458 70 6 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 70 7 } } PUNCT cana-1458 70 8 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 70 9 { { PUNCT cana-1458 70 10 ( ( PUNCT cana-1458 70 11 𝐺 𝐺 NOUN cana-1458 70 12 , , PUNCT cana-1458 70 13 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 70 14 } } PUNCT cana-1458 70 15 ∈ ∈ PROPN cana-1458 70 16 οΏ½ οΏ½ PROPN cana-1458 70 17 Μƒ Μƒ PROPN cana-1458 70 18 οΏ½ οΏ½ NOUN cana-1458 70 19 1 1 NUM cana-1458 70 20 and and CCONJ cana-1458 70 21 { { PUNCT cana-1458 70 22 ( ( PUNCT cana-1458 70 23 𝐹 𝐹 PROPN cana-1458 70 24 , , PUNCT cana-1458 70 25 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 70 26 } } PUNCT cana-1458 70 27 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 70 28 { { PUNCT cana-1458 70 29 ( ( PUNCT cana-1458 70 30 𝐺 𝐺 NOUN cana-1458 70 31 , , PUNCT cana-1458 70 32 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 70 33 } } PUNCT cana-1458 70 34 ∈ ∈ PROPN cana-1458 70 35 οΏ½ οΏ½ PROPN cana-1458 70 36 Μƒ Μƒ PROPN cana-1458 70 37 οΏ½ οΏ½ NOUN cana-1458 70 38 2 2 NUM cana-1458 70 39 . . PUNCT cana-1458 70 40 hence hence ADV cana-1458 70 41 { { PUNCT cana-1458 70 42 ( ( PUNCT cana-1458 70 43 𝐹 𝐹 PROPN cana-1458 70 44 , , PUNCT cana-1458 70 45 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 70 46 } } PUNCT cana-1458 70 47 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 70 48 { { PUNCT cana-1458 70 49 ( ( PUNCT cana-1458 70 50 𝐺 𝐺 NOUN cana-1458 70 51 , , PUNCT cana-1458 70 52 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 70 53 } } PUNCT cana-1458 70 54 ∈ ∈ PROPN cana-1458 70 55 οΏ½ οΏ½ PROPN cana-1458 70 56 Μƒ Μƒ PROPN cana-1458 70 57 οΏ½ οΏ½ PROPN cana-1458 70 58 1 1 NUM cana-1458 70 59 βˆ©Μƒ βˆ©Μƒ SYM cana-1458 70 60 οΏ½ οΏ½ PROPN cana-1458 70 61 Μƒ Μƒ PROPN cana-1458 70 62 οΏ½ οΏ½ NOUN cana-1458 70 63 2 2 NUM cana-1458 70 64 . . PUNCT cana-1458 70 65 communications communication NOUN cana-1458 70 66 on on ADP cana-1458 70 67 applied apply VERB cana-1458 70 68 nonlinear nonlinear ADJ cana-1458 70 69 analysis analysis NOUN cana-1458 70 70 issn issn NOUN cana-1458 70 71 : : PUNCT cana-1458 70 72 1074 1074 NUM cana-1458 70 73 - - PUNCT cana-1458 70 74 133x 133x NUM cana-1458 70 75 vol vol NOUN cana-1458 70 76 31 31 NUM cana-1458 70 77 no no NOUN cana-1458 70 78 . . PUNCT cana-1458 71 1 8s 8s PROPN cana-1458 71 2 ( ( PUNCT cana-1458 71 3 2024 2024 NUM cana-1458 71 4 ) ) PUNCT cana-1458 71 5 90 90 NUM cana-1458 71 6 https://internationalpubls.com https://internationalpubls.com X cana-1458 71 7 ( ( PUNCT cana-1458 71 8 iii iii NOUN cana-1458 71 9 ) ) PUNCT cana-1458 71 10 let let VERB cana-1458 71 11 { { PUNCT cana-1458 71 12 { { PUNCT cana-1458 71 13 ( ( PUNCT cana-1458 71 14 πΉπœ† πΉπœ† NOUN cana-1458 71 15 , , PUNCT cana-1458 71 16 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 71 17 } } PUNCT cana-1458 71 18 , , PUNCT cana-1458 71 19 πœ† πœ† PROPN cana-1458 71 20 ∈ ∈ PROPN cana-1458 71 21 Ξ» Ξ» PROPN cana-1458 71 22 } } PUNCT cana-1458 71 23 be be VERB cana-1458 71 24 a a DET cana-1458 71 25 family family NOUN cana-1458 71 26 of of ADP cana-1458 71 27 fuzzy fuzzy ADJ cana-1458 71 28 soft soft ADJ cana-1458 71 29 sequential sequential ADJ cana-1458 71 30 sets set NOUN cana-1458 71 31 in in ADP cana-1458 71 32 οΏ½ οΏ½ PROPN cana-1458 71 33 Μƒ Μƒ NOUN cana-1458 71 34 οΏ½ οΏ½ PROPN cana-1458 71 35 1 1 NUM cana-1458 71 36 βˆ©Μƒ βˆ©Μƒ SYM cana-1458 71 37 οΏ½ οΏ½ PROPN cana-1458 71 38 Μƒ Μƒ PROPN cana-1458 71 39 οΏ½ οΏ½ NOUN cana-1458 71 40 2 2 NUM cana-1458 71 41 . . PUNCT cana-1458 72 1 then then ADV cana-1458 72 2 { { PUNCT cana-1458 72 3 ( ( PUNCT cana-1458 72 4 πΉπœ† πΉπœ† NOUN cana-1458 72 5 , , PUNCT cana-1458 72 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 72 7 } } PUNCT cana-1458 72 8 ∈ ∈ PROPN cana-1458 72 9 οΏ½ οΏ½ PROPN cana-1458 72 10 Μƒ Μƒ PROPN cana-1458 72 11 οΏ½ οΏ½ NOUN cana-1458 72 12 1 1 NUM cana-1458 72 13 and and CCONJ cana-1458 72 14 { { PUNCT cana-1458 72 15 ( ( PUNCT cana-1458 72 16 πΉπœ† πΉπœ† NOUN cana-1458 72 17 , , PUNCT cana-1458 72 18 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 72 19 } } PUNCT cana-1458 72 20 ∈ ∈ PROPN cana-1458 72 21 οΏ½ οΏ½ PROPN cana-1458 72 22 Μƒ Μƒ PROPN cana-1458 72 23 οΏ½ οΏ½ NOUN cana-1458 72 24 2 2 NUM cana-1458 72 25 βˆ€πœ† βˆ€πœ† NOUN cana-1458 72 26 ∈ ∈ PROPN cana-1458 72 27 Ξ» Ξ» PROPN cana-1458 72 28 . . PUNCT cana-1458 73 1 therefore therefore ADV cana-1458 73 2 { { PUNCT cana-1458 73 3 ( ( PUNCT cana-1458 73 4 πΉπœ† πΉπœ† PROPN cana-1458 73 5 , , PUNCT cana-1458 73 6 𝐴)𝑛}πœ†βˆˆΞ» 𝐴)𝑛}πœ†βˆˆΞ» PROPN cana-1458 73 7 βˆͺΜƒ βˆͺΜƒ PROPN cana-1458 73 8 ∈ ∈ PROPN cana-1458 73 9 οΏ½ οΏ½ PROPN cana-1458 73 10 Μƒ Μƒ PROPN cana-1458 73 11 οΏ½ οΏ½ NOUN cana-1458 73 12 1 1 NUM cana-1458 73 13 and and CCONJ cana-1458 73 14 { { PUNCT cana-1458 73 15 ( ( PUNCT cana-1458 73 16 πΉπœ† πΉπœ† PROPN cana-1458 73 17 , , PUNCT cana-1458 73 18 𝐴)𝑛}πœ†βˆˆΞ» 𝐴)𝑛}πœ†βˆˆΞ» PROPN cana-1458 73 19 βˆͺΜƒ βˆͺΜƒ PROPN cana-1458 73 20 ∈ ∈ PROPN cana-1458 73 21 οΏ½ οΏ½ PROPN cana-1458 73 22 Μƒ Μƒ PROPN cana-1458 73 23 οΏ½ οΏ½ NOUN cana-1458 73 24 2 2 NUM cana-1458 73 25 . . PUNCT cana-1458 73 26 hence hence ADV cana-1458 73 27 { { PUNCT cana-1458 73 28 ( ( PUNCT cana-1458 73 29 πΉπœ† πΉπœ† PROPN cana-1458 73 30 , , PUNCT cana-1458 73 31 𝐴)𝑛}πœ†βˆˆΞ» 𝐴)𝑛}πœ†βˆˆΞ» PROPN cana-1458 73 32 βˆͺΜƒ βˆͺΜƒ PROPN cana-1458 73 33 ∈ ∈ PROPN cana-1458 73 34 οΏ½ οΏ½ PROPN cana-1458 73 35 Μƒ Μƒ PROPN cana-1458 73 36 οΏ½ οΏ½ PROPN cana-1458 73 37 1 1 NUM cana-1458 73 38 βˆ©Μƒ βˆ©Μƒ SYM cana-1458 73 39 οΏ½ οΏ½ PROPN cana-1458 73 40 Μƒ Μƒ PROPN cana-1458 73 41 οΏ½ οΏ½ NOUN cana-1458 73 42 2 2 NUM cana-1458 73 43 . . PUNCT cana-1458 73 44 remark remark PROPN cana-1458 73 45 3.10 3.10 NUM cana-1458 73 46 . . PUNCT cana-1458 74 1 let let VERB cana-1458 74 2 ( ( PUNCT cana-1458 74 3 π‘ˆ π‘ˆ PROPN cana-1458 74 4 , , PUNCT cana-1458 74 5 οΏ½ οΏ½ PROPN cana-1458 74 6 Μƒ Μƒ NOUN cana-1458 74 7 οΏ½ οΏ½ NOUN cana-1458 74 8 1 1 NUM cana-1458 74 9 , , PUNCT cana-1458 74 10 𝐸 𝐸 PROPN cana-1458 74 11 ) ) PUNCT cana-1458 74 12 and and CCONJ cana-1458 74 13 ( ( PUNCT cana-1458 74 14 π‘ˆ π‘ˆ PROPN cana-1458 74 15 , , PUNCT cana-1458 74 16 οΏ½ οΏ½ PROPN cana-1458 74 17 Μƒ Μƒ NOUN cana-1458 74 18 οΏ½ οΏ½ NOUN cana-1458 74 19 2 2 NUM cana-1458 74 20 , , PUNCT cana-1458 74 21 𝐸 𝐸 PROPN cana-1458 74 22 ) ) PUNCT cana-1458 74 23 be be VERB cana-1458 74 24 two two NUM cana-1458 74 25 fuzzy fuzzy ADJ cana-1458 74 26 soft soft ADJ cana-1458 74 27 sequential sequential ADJ cana-1458 74 28 topological topological ADJ cana-1458 74 29 spaces space NOUN cana-1458 74 30 over over ADP cana-1458 74 31 π‘ˆ π‘ˆ PROPN cana-1458 74 32 , , PUNCT cana-1458 74 33 then then ADV cana-1458 74 34 ( ( PUNCT cana-1458 74 35 π‘ˆ π‘ˆ PROPN cana-1458 74 36 , , PUNCT cana-1458 74 37 οΏ½ οΏ½ PROPN cana-1458 74 38 Μƒ Μƒ PROPN cana-1458 74 39 οΏ½ οΏ½ PROPN cana-1458 74 40 1 1 NUM cana-1458 74 41 βˆͺΜƒ βˆͺΜƒ PROPN cana-1458 74 42 οΏ½ οΏ½ PROPN cana-1458 74 43 Μƒ Μƒ PROPN cana-1458 74 44 οΏ½ οΏ½ NOUN cana-1458 74 45 2 2 NUM cana-1458 74 46 , , PUNCT cana-1458 74 47 𝐸 𝐸 PROPN cana-1458 74 48 ) ) PUNCT cana-1458 74 49 need need AUX cana-1458 74 50 not not PART cana-1458 74 51 be be AUX cana-1458 74 52 a a DET cana-1458 74 53 fuzzy fuzzy ADJ cana-1458 74 54 soft soft ADJ cana-1458 74 55 sequential sequential ADJ cana-1458 74 56 topological topological ADJ cana-1458 74 57 space space NOUN cana-1458 74 58 over over ADP cana-1458 74 59 π‘ˆ. π‘ˆ. PROPN cana-1458 74 60 proposition proposition NOUN cana-1458 74 61 3.11 3.11 NUM cana-1458 74 62 . . PUNCT cana-1458 75 1 if if SCONJ cana-1458 75 2 οΏ½ οΏ½ PROPN cana-1458 75 3 Μƒ Μƒ PROPN cana-1458 75 4 οΏ½ οΏ½ PROPN cana-1458 75 5 is be AUX cana-1458 75 6 a a DET cana-1458 75 7 fuzzy fuzzy ADJ cana-1458 75 8 soft soft ADJ cana-1458 75 9 topology topology NOUN cana-1458 75 10 on on ADP cana-1458 75 11 π‘ˆ π‘ˆ PROPN cana-1458 75 12 , , PUNCT cana-1458 75 13 then then ADV cana-1458 75 14 οΏ½ οΏ½ PROPN cana-1458 75 15 Μƒ Μƒ PROPN cana-1458 75 16 οΏ½ οΏ½ PROPN cana-1458 75 17 β„• β„• PROPN cana-1458 75 18 forms form VERB cana-1458 75 19 a a DET cana-1458 75 20 fuzzy fuzzy ADJ cana-1458 75 21 soft soft ADJ cana-1458 75 22 sequential sequential ADJ cana-1458 75 23 topology topology NOUN cana-1458 75 24 on on ADP cana-1458 75 25 π‘ˆ. π‘ˆ. PROPN cana-1458 75 26 proof proof NOUN cana-1458 75 27 . . PUNCT cana-1458 76 1 let let VERB cana-1458 76 2 οΏ½ οΏ½ PROPN cana-1458 76 3 Μƒ Μƒ PROPN cana-1458 76 4 οΏ½ οΏ½ PROPN cana-1458 76 5 be be AUX cana-1458 76 6 a a DET cana-1458 76 7 fuzzy fuzzy ADJ cana-1458 76 8 soft soft ADJ cana-1458 76 9 topology topology NOUN cana-1458 76 10 on on ADP cana-1458 76 11 π‘ˆ. π‘ˆ. PROPN cana-1458 76 12 ( ( PUNCT cana-1458 76 13 i i NOUN cana-1458 76 14 ) ) PUNCT cana-1458 76 15 βˆ…Μƒ βˆ…Μƒ PROPN cana-1458 76 16 , , PUNCT cana-1458 76 17 οΏ½ οΏ½ PROPN cana-1458 76 18 Μƒ Μƒ PROPN cana-1458 76 19 οΏ½ οΏ½ PROPN cana-1458 76 20 ∈ ∈ PROPN cana-1458 76 21 οΏ½ οΏ½ PROPN cana-1458 76 22 Μƒ Μƒ PROPN cana-1458 76 23 οΏ½ οΏ½ PROPN cana-1458 76 24 β„• β„• PROPN cana-1458 76 25 , , PUNCT cana-1458 76 26 since since SCONJ cana-1458 76 27 βˆ…Μƒ βˆ…Μƒ NOUN cana-1458 76 28 , , PUNCT cana-1458 76 29 π‘ˆ π‘ˆ PROPN cana-1458 76 30 ∈ ∈ PROPN cana-1458 76 31 οΏ½ οΏ½ PROPN cana-1458 76 32 Μƒ Μƒ PROPN cana-1458 76 33 οΏ½ οΏ½ PROPN cana-1458 76 34 . . PUNCT cana-1458 77 1 ( ( PUNCT cana-1458 77 2 ii ii NOUN cana-1458 77 3 ) ) PUNCT cana-1458 77 4 let let VERB cana-1458 77 5 { { PUNCT cana-1458 77 6 ( ( PUNCT cana-1458 77 7 𝐹 𝐹 PROPN cana-1458 77 8 , , PUNCT cana-1458 77 9 𝐴)𝑛 𝐴)𝑛 PROPN cana-1458 77 10 } } PUNCT cana-1458 77 11 , , PUNCT cana-1458 77 12 { { PUNCT cana-1458 77 13 ( ( PUNCT cana-1458 77 14 𝐺 𝐺 NOUN cana-1458 77 15 , , PUNCT cana-1458 77 16 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 77 17 } } PUNCT cana-1458 77 18 ∈ ∈ PROPN cana-1458 77 19 οΏ½ οΏ½ PROPN cana-1458 77 20 Μƒ Μƒ PROPN cana-1458 77 21 οΏ½ οΏ½ PROPN cana-1458 77 22 β„•. β„•. PROPN cana-1458 77 23 then then ADV cana-1458 77 24 ( ( PUNCT cana-1458 77 25 𝐹 𝐹 PROPN cana-1458 77 26 , , PUNCT cana-1458 77 27 𝐴)𝑛,(𝐺 𝐴)𝑛,(𝐺 PROPN cana-1458 77 28 , , PUNCT cana-1458 77 29 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 77 30 ∈ ∈ PROPN cana-1458 77 31 οΏ½ οΏ½ PROPN cana-1458 77 32 Μƒ Μƒ PROPN cana-1458 77 33 οΏ½ οΏ½ PROPN cana-1458 77 34 for for ADP cana-1458 77 35 all all DET cana-1458 77 36 𝑛 𝑛 DET cana-1458 77 37 ∈ ∈ PROPN cana-1458 77 38 β„•. β„•. PROPN cana-1458 77 39 thus thus ADV cana-1458 77 40 ( ( PUNCT cana-1458 77 41 𝐹 𝐹 PROPN cana-1458 77 42 , , PUNCT cana-1458 77 43 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 77 44 βˆ©Μƒ βˆ©Μƒ SYM cana-1458 78 1 ( ( PUNCT cana-1458 78 2 𝐺 𝐺 PROPN cana-1458 78 3 , , PUNCT cana-1458 78 4 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 78 5 ∈ ∈ PROPN cana-1458 78 6 οΏ½ οΏ½ PROPN cana-1458 78 7 Μƒ Μƒ PROPN cana-1458 78 8 οΏ½ οΏ½ PROPN cana-1458 78 9 for for ADP cana-1458 78 10 all all DET cana-1458 78 11 𝑛 𝑛 DET cana-1458 78 12 ∈ ∈ PROPN cana-1458 78 13 β„•. β„•. PROPN cana-1458 78 14 thus thus ADV cana-1458 78 15 { { PUNCT cana-1458 78 16 ( ( PUNCT cana-1458 78 17 𝐹 𝐹 PROPN cana-1458 78 18 , , PUNCT cana-1458 78 19 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 78 20 } } PUNCT cana-1458 78 21 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 78 22 { { PUNCT cana-1458 78 23 ( ( PUNCT cana-1458 78 24 𝐺 𝐺 NOUN cana-1458 78 25 , , PUNCT cana-1458 78 26 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 78 27 } } PUNCT cana-1458 78 28 ∈ ∈ PROPN cana-1458 78 29 οΏ½ οΏ½ PROPN cana-1458 78 30 Μƒ Μƒ PROPN cana-1458 78 31 οΏ½ οΏ½ PROPN cana-1458 78 32 β„•. β„•. PROPN cana-1458 78 33 ( ( PUNCT cana-1458 78 34 iii)let iii)let NOUN cana-1458 78 35 { { PUNCT cana-1458 78 36 { { PUNCT cana-1458 78 37 ( ( PUNCT cana-1458 78 38 πΉπœ† πΉπœ† NOUN cana-1458 78 39 , , PUNCT cana-1458 78 40 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 78 41 } } PUNCT cana-1458 78 42 ∈ ∈ PROPN cana-1458 78 43 οΏ½ οΏ½ PROPN cana-1458 78 44 Μƒ Μƒ PROPN cana-1458 78 45 οΏ½ οΏ½ PROPN cana-1458 78 46 , , PUNCT cana-1458 78 47 πœ† πœ† PROPN cana-1458 78 48 ∈ ∈ PROPN cana-1458 78 49 Ξ» Ξ» PROPN cana-1458 78 50 } } PUNCT cana-1458 78 51 be be VERB cana-1458 78 52 a a DET cana-1458 78 53 family family NOUN cana-1458 78 54 of of ADP cana-1458 78 55 fuzzy fuzzy ADJ cana-1458 78 56 soft soft ADJ cana-1458 78 57 sequential sequential ADJ cana-1458 78 58 sets set NOUN cana-1458 78 59 in in ADP cana-1458 78 60 οΏ½ οΏ½ PROPN cana-1458 78 61 Μƒ Μƒ PROPN cana-1458 78 62 οΏ½ οΏ½ PROPN cana-1458 78 63 β„•. β„•. PROPN cana-1458 78 64 then then ADV cana-1458 78 65 for for ADP cana-1458 78 66 each each DET cana-1458 78 67 𝑛 𝑛 PRON cana-1458 78 68 ∈ ∈ PROPN cana-1458 78 69 β„• β„• PROPN cana-1458 78 70 , , PUNCT cana-1458 78 71 ( ( PUNCT cana-1458 78 72 𝐹 𝐹 PROPN cana-1458 78 73 , , PUNCT cana-1458 78 74 𝐴)πœ† 𝐴)πœ† PROPN cana-1458 78 75 ∈ ∈ PROPN cana-1458 78 76 οΏ½ οΏ½ PROPN cana-1458 78 77 Μƒ Μƒ PROPN cana-1458 78 78 οΏ½ οΏ½ PROPN cana-1458 78 79 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-1458 78 80 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ VERB cana-1458 78 81 πœ† πœ† DET cana-1458 78 82 ∈ ∈ PROPN cana-1458 78 83 Ξ» Ξ» PROPN cana-1458 78 84 . . PUNCT cana-1458 79 1 thus thus ADV cana-1458 79 2 πœ†βˆˆΞ» πœ†βˆˆΞ» PROPN cana-1458 79 3 βˆͺΜƒ βˆͺΜƒ PROPN cana-1458 79 4 ( ( PUNCT cana-1458 79 5 𝐹 𝐹 PROPN cana-1458 79 6 , , PUNCT cana-1458 79 7 𝐴)πœ† 𝐴)πœ† PROPN cana-1458 79 8 ∈ ∈ PROPN cana-1458 79 9 οΏ½ οΏ½ PROPN cana-1458 79 10 Μƒ Μƒ PROPN cana-1458 79 11 οΏ½ οΏ½ PROPN cana-1458 79 12 . . PUNCT cana-1458 80 1 hence hence ADV cana-1458 80 2 { { PUNCT cana-1458 80 3 ( ( PUNCT cana-1458 80 4 πΉπœ† πΉπœ† PROPN cana-1458 80 5 , , PUNCT cana-1458 80 6 𝐴)𝑛}πœ†βˆˆΞ» 𝐴)𝑛}πœ†βˆˆΞ» PROPN cana-1458 80 7 βˆͺΜƒ βˆͺΜƒ PROPN cana-1458 80 8 ∈ ∈ PROPN cana-1458 80 9 οΏ½ οΏ½ PROPN cana-1458 80 10 Μƒ Μƒ PROPN cana-1458 80 11 οΏ½ οΏ½ PROPN cana-1458 80 12 β„•. β„•. PROPN cana-1458 80 13 proposition proposition NOUN cana-1458 80 14 3.12 3.12 NUM cana-1458 80 15 . . PUNCT cana-1458 81 1 any any DET cana-1458 81 2 fuzzy fuzzy ADJ cana-1458 81 3 soft soft ADJ cana-1458 81 4 sequential sequential ADJ cana-1458 81 5 topological topological ADJ cana-1458 81 6 space space NOUN cana-1458 81 7 induces induce VERB cana-1458 81 8 fuzzy fuzzy ADJ cana-1458 81 9 soft soft ADJ cana-1458 81 10 topological topological ADJ cana-1458 81 11 space space NOUN cana-1458 81 12 . . PUNCT cana-1458 82 1 proof proof NOUN cana-1458 82 2 . . PUNCT cana-1458 83 1 let let VERB cana-1458 83 2 οΏ½ οΏ½ PROPN cana-1458 83 3 Μƒ Μƒ PROPN cana-1458 83 4 οΏ½ οΏ½ PROPN cana-1458 83 5 be be AUX cana-1458 83 6 a a DET cana-1458 83 7 fuzzy fuzzy ADJ cana-1458 83 8 soft soft ADJ cana-1458 83 9 sequential sequential ADJ cana-1458 83 10 topology topology NOUN cana-1458 83 11 , , PUNCT cana-1458 83 12 οΏ½ οΏ½ PROPN cana-1458 83 13 Μƒ Μƒ PROPN cana-1458 83 14 οΏ½ οΏ½ NOUN cana-1458 83 15 1 1 NUM cana-1458 83 16 be be VERB cana-1458 83 17 a a DET cana-1458 83 18 fuzzy fuzzy ADJ cana-1458 83 19 soft soft ADJ cana-1458 83 20 topology topology NOUN cana-1458 83 21 and and CCONJ cana-1458 83 22 𝑛 𝑛 DET cana-1458 83 23 ∈ ∈ PROPN cana-1458 83 24 β„• β„• PROPN cana-1458 83 25 , , PUNCT cana-1458 83 26 the the DET cana-1458 83 27 set set NOUN cana-1458 83 28 of of ADP cana-1458 83 29 all all DET cana-1458 83 30 natural natural ADJ cana-1458 83 31 numbers number NOUN cana-1458 83 32 . . PUNCT cana-1458 84 1 then then ADV cana-1458 84 2 ( ( PUNCT cana-1458 84 3 i i NOUN cana-1458 84 4 ) ) PUNCT cana-1458 84 5 βˆ…Μƒ βˆ…Μƒ NOUN cana-1458 84 6 , , PUNCT cana-1458 84 7 π‘ˆ π‘ˆ PROPN cana-1458 84 8 ∈ ∈ PROPN cana-1458 84 9 οΏ½ οΏ½ PROPN cana-1458 84 10 Μƒ Μƒ NOUN cana-1458 84 11 οΏ½ οΏ½ NOUN cana-1458 84 12 1 1 NUM cana-1458 84 13 , , PUNCT cana-1458 84 14 since since SCONJ cana-1458 84 15 βˆ…Μƒ βˆ…Μƒ NOUN cana-1458 84 16 , , PUNCT cana-1458 84 17 οΏ½ οΏ½ PROPN cana-1458 84 18 Μƒ Μƒ PROPN cana-1458 84 19 οΏ½ οΏ½ PROPN cana-1458 84 20 ∈ ∈ PROPN cana-1458 84 21 οΏ½ οΏ½ PROPN cana-1458 84 22 Μƒ Μƒ PROPN cana-1458 84 23 οΏ½ οΏ½ PROPN cana-1458 84 24 . . PUNCT cana-1458 84 25 ( ( PUNCT cana-1458 84 26 ii ii NOUN cana-1458 84 27 ) ) PUNCT cana-1458 84 28 let let VERB cana-1458 84 29 ( ( PUNCT cana-1458 84 30 𝐹 𝐹 PROPN cana-1458 84 31 , , PUNCT cana-1458 84 32 𝐴 𝐴 PROPN cana-1458 84 33 ) ) PUNCT cana-1458 84 34 , , PUNCT cana-1458 84 35 ( ( PUNCT cana-1458 84 36 𝐺 𝐺 NOUN cana-1458 84 37 , , PUNCT cana-1458 84 38 𝐡 𝐡 PROPN cana-1458 84 39 ) ) PUNCT cana-1458 84 40 ∈ ∈ PROPN cana-1458 84 41 οΏ½ οΏ½ PROPN cana-1458 84 42 Μƒ Μƒ PROPN cana-1458 84 43 οΏ½ οΏ½ NOUN cana-1458 84 44 1 1 NUM cana-1458 84 45 . . PUNCT cana-1458 85 1 then then ADV cana-1458 85 2 there there PRON cana-1458 85 3 exists exist VERB cana-1458 85 4 fuzzy fuzzy ADJ cana-1458 85 5 soft soft ADJ cana-1458 85 6 sequential sequential ADJ cana-1458 85 7 sets set NOUN cana-1458 85 8 { { PUNCT cana-1458 85 9 ( ( PUNCT cana-1458 85 10 𝐹 𝐹 PROPN cana-1458 85 11 , , PUNCT cana-1458 85 12 𝐴)𝑛},{(𝐺 𝐴)𝑛},{(𝐺 PROPN cana-1458 85 13 , , PUNCT cana-1458 85 14 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 85 15 } } PUNCT cana-1458 85 16 ∈ ∈ PROPN cana-1458 85 17 οΏ½ οΏ½ PROPN cana-1458 85 18 Μƒ Μƒ NOUN cana-1458 85 19 οΏ½ οΏ½ PROPN cana-1458 85 20 such such ADJ cana-1458 85 21 that that SCONJ cana-1458 85 22 ( ( PUNCT cana-1458 85 23 𝐹 𝐹 PROPN cana-1458 85 24 , , PUNCT cana-1458 85 25 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 86 1 = = SYM cana-1458 86 2 ( ( PUNCT cana-1458 86 3 𝐹 𝐹 PROPN cana-1458 86 4 , , PUNCT cana-1458 86 5 𝐴 𝐴 PROPN cana-1458 86 6 ) ) PUNCT cana-1458 86 7 and and CCONJ cana-1458 86 8 ( ( PUNCT cana-1458 86 9 𝐺 𝐺 PROPN cana-1458 86 10 , , PUNCT cana-1458 86 11 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 86 12 = = SYM cana-1458 86 13 ( ( PUNCT cana-1458 86 14 𝐺 𝐺 PROPN cana-1458 86 15 , , PUNCT cana-1458 86 16 𝐡 𝐡 PROPN cana-1458 86 17 ) ) PUNCT cana-1458 86 18 . . PUNCT cana-1458 87 1 since since SCONJ cana-1458 87 2 { { PUNCT cana-1458 87 3 ( ( PUNCT cana-1458 87 4 𝐹 𝐹 PROPN cana-1458 87 5 , , PUNCT cana-1458 87 6 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 87 7 } } PUNCT cana-1458 87 8 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 87 9 { { PUNCT cana-1458 87 10 ( ( PUNCT cana-1458 87 11 𝐺 𝐺 NOUN cana-1458 87 12 , , PUNCT cana-1458 87 13 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 87 14 } } PUNCT cana-1458 87 15 ∈ ∈ PROPN cana-1458 87 16 οΏ½ οΏ½ PROPN cana-1458 87 17 Μƒ Μƒ PROPN cana-1458 87 18 οΏ½ οΏ½ PROPN cana-1458 87 19 implies imply VERB cana-1458 87 20 ( ( PUNCT cana-1458 87 21 𝐹 𝐹 PROPN cana-1458 87 22 , , PUNCT cana-1458 87 23 𝐴 𝐴 PROPN cana-1458 87 24 ) ) PUNCT cana-1458 87 25 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 87 26 ( ( PUNCT cana-1458 87 27 𝐺 𝐺 PROPN cana-1458 87 28 , , PUNCT cana-1458 87 29 𝐡 𝐡 PROPN cana-1458 87 30 ) ) PUNCT cana-1458 87 31 ∈ ∈ PROPN cana-1458 87 32 οΏ½ οΏ½ PROPN cana-1458 87 33 Μƒ Μƒ PROPN cana-1458 87 34 οΏ½ οΏ½ NOUN cana-1458 87 35 1 1 NUM cana-1458 87 36 . . PUNCT cana-1458 88 1 ( ( PUNCT cana-1458 88 2 iii)let iii)let NOUN cana-1458 88 3 { { PUNCT cana-1458 88 4 ( ( PUNCT cana-1458 88 5 𝐹 𝐹 PROPN cana-1458 88 6 , , PUNCT cana-1458 88 7 𝐴)πœ† 𝐴)πœ† X cana-1458 88 8 βˆ• βˆ• PUNCT cana-1458 88 9 πœ† πœ† PRON cana-1458 88 10 ∈ ∈ PROPN cana-1458 88 11 Ξ» Ξ» PROPN cana-1458 88 12 } } PUNCT cana-1458 88 13 be be VERB cana-1458 88 14 a a DET cana-1458 88 15 family family NOUN cana-1458 88 16 of of ADP cana-1458 88 17 fuzzy fuzzy ADJ cana-1458 88 18 soft soft ADJ cana-1458 88 19 open open ADJ cana-1458 88 20 sets set NOUN cana-1458 88 21 . . PUNCT cana-1458 89 1 then then ADV cana-1458 89 2 for for ADP cana-1458 89 3 each each DET cana-1458 89 4 πœ† πœ† PRON cana-1458 89 5 ∈ ∈ PROPN cana-1458 89 6 Ξ» Ξ» PROPN cana-1458 89 7 , , PUNCT cana-1458 89 8 there there PRON cana-1458 89 9 exists exist VERB cana-1458 89 10 a a DET cana-1458 89 11 fuzzy fuzzy ADJ cana-1458 89 12 soft soft ADJ cana-1458 89 13 sequential sequential ADJ cana-1458 89 14 set set NOUN cana-1458 89 15 { { PUNCT cana-1458 89 16 { { PUNCT cana-1458 89 17 ( ( PUNCT cana-1458 89 18 πΉπœ† πΉπœ† NOUN cana-1458 89 19 , , PUNCT cana-1458 89 20 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 89 21 } } PUNCT cana-1458 89 22 βˆ• βˆ• PUNCT cana-1458 89 23 πœ† πœ† DET cana-1458 89 24 ∈ ∈ PROPN cana-1458 89 25 Ξ» Ξ» PROPN cana-1458 89 26 } } PUNCT cana-1458 89 27 ∈ ∈ PROPN cana-1458 89 28 οΏ½ οΏ½ PROPN cana-1458 89 29 Μƒ Μƒ PROPN cana-1458 89 30 οΏ½ οΏ½ PROPN cana-1458 89 31 having have VERB cana-1458 89 32 nth nth PROPN cana-1458 89 33 component component NOUN cana-1458 89 34 ( ( PUNCT cana-1458 89 35 𝐹 𝐹 PROPN cana-1458 89 36 , , PUNCT cana-1458 89 37 𝐴)πœ†. 𝐴)πœ†. PROPN cana-1458 89 38 since since SCONJ cana-1458 89 39 { { PUNCT cana-1458 89 40 ( ( PUNCT cana-1458 89 41 πΉπœ† πΉπœ† PROPN cana-1458 89 42 , , PUNCT cana-1458 89 43 𝐴)𝑛}πœ†βˆˆΞ» 𝐴)𝑛}πœ†βˆˆΞ» PROPN cana-1458 89 44 βˆͺΜƒ βˆͺΜƒ PROPN cana-1458 89 45 ∈ ∈ PROPN cana-1458 89 46 οΏ½ οΏ½ PROPN cana-1458 89 47 Μƒ Μƒ PROPN cana-1458 89 48 οΏ½ οΏ½ PROPN cana-1458 89 49 , , PUNCT cana-1458 89 50 πœ†βˆˆΞ» πœ†βˆˆΞ» PROPN cana-1458 89 51 βˆͺΜƒ βˆͺΜƒ PROPN cana-1458 89 52 ( ( PUNCT cana-1458 89 53 𝐹 𝐹 PROPN cana-1458 89 54 , , PUNCT cana-1458 89 55 𝐴)πœ† 𝐴)πœ† PROPN cana-1458 89 56 ∈ ∈ PROPN cana-1458 89 57 οΏ½ οΏ½ PROPN cana-1458 89 58 Μƒ Μƒ PROPN cana-1458 89 59 οΏ½ οΏ½ NOUN cana-1458 89 60 1 1 NUM cana-1458 89 61 . . PUNCT cana-1458 89 62 definition definition NOUN cana-1458 89 63 3.13 3.13 NUM cana-1458 89 64 . . PUNCT cana-1458 90 1 ( ( PUNCT cana-1458 90 2 π‘ˆ π‘ˆ PROPN cana-1458 90 3 , , PUNCT cana-1458 90 4 οΏ½ οΏ½ PROPN cana-1458 90 5 Μƒ Μƒ PROPN cana-1458 90 6 οΏ½ οΏ½ PROPN cana-1458 90 7 ) ) PUNCT cana-1458 90 8 in in ADP cana-1458 90 9 the the DET cana-1458 90 10 proposition proposition NOUN cana-1458 90 11 3.12 3.12 NUM cana-1458 90 12 , , PUNCT cana-1458 90 13 is be AUX cana-1458 90 14 called call VERB cana-1458 90 15 the the DET cana-1458 90 16 nth nth NOUN cana-1458 90 17 component component NOUN cana-1458 90 18 fuzzy fuzzy ADJ cana-1458 90 19 soft soft ADJ cana-1458 90 20 topological topological ADJ cana-1458 90 21 space space NOUN cana-1458 90 22 of of ADP cana-1458 90 23 the the DET cana-1458 90 24 fuzzy fuzzy ADJ cana-1458 90 25 soft soft ADJ cana-1458 90 26 sequential sequential ADJ cana-1458 90 27 topological topological ADJ cana-1458 90 28 space space NOUN cana-1458 90 29 ( ( PUNCT cana-1458 90 30 π‘ˆ π‘ˆ PROPN cana-1458 90 31 , , PUNCT cana-1458 90 32 οΏ½ οΏ½ PROPN cana-1458 90 33 Μƒ Μƒ PROPN cana-1458 90 34 οΏ½ οΏ½ PROPN cana-1458 90 35 , , PUNCT cana-1458 90 36 𝐸 𝐸 PROPN cana-1458 90 37 ) ) PUNCT cana-1458 90 38 . . PUNCT cana-1458 91 1 proposition proposition NOUN cana-1458 91 2 3.14 3.14 NUM cana-1458 91 3 . . PUNCT cana-1458 92 1 let let AUX cana-1458 92 2 { { PUNCT cana-1458 92 3 ( ( PUNCT cana-1458 92 4 𝐹 𝐹 PROPN cana-1458 92 5 , , PUNCT cana-1458 92 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 92 7 } } PUNCT cana-1458 92 8 be be AUX cana-1458 92 9 a a DET cana-1458 92 10 fuzzy fuzzy ADJ cana-1458 92 11 soft soft ADJ cana-1458 92 12 sequential sequential ADJ cana-1458 92 13 open open ADJ cana-1458 92 14 ( ( PUNCT cana-1458 92 15 closed closed ADJ cana-1458 92 16 ) ) PUNCT cana-1458 92 17 set set VERB cana-1458 92 18 in in ADP cana-1458 92 19 fuzzy fuzzy ADJ cana-1458 92 20 soft soft ADJ cana-1458 92 21 sequential sequential ADJ cana-1458 92 22 topological topological ADJ cana-1458 92 23 space space NOUN cana-1458 92 24 ( ( PUNCT cana-1458 92 25 π‘ˆ π‘ˆ PROPN cana-1458 92 26 , , PUNCT cana-1458 92 27 οΏ½ οΏ½ PROPN cana-1458 92 28 Μƒ Μƒ PROPN cana-1458 92 29 οΏ½ οΏ½ PROPN cana-1458 92 30 , , PUNCT cana-1458 92 31 𝐸 𝐸 PROPN cana-1458 92 32 ) ) PUNCT cana-1458 92 33 . . PUNCT cana-1458 93 1 then then ADV cana-1458 93 2 for for ADP cana-1458 93 3 each each DET cana-1458 93 4 𝑛 𝑛 PRON cana-1458 93 5 ∈ ∈ PROPN cana-1458 93 6 β„• β„• PROPN cana-1458 93 7 , , PUNCT cana-1458 93 8 ( ( PUNCT cana-1458 93 9 𝐹 𝐹 PROPN cana-1458 93 10 , , PUNCT cana-1458 93 11 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 93 12 is be AUX cana-1458 93 13 fuzzy fuzzy ADJ cana-1458 93 14 soft soft ADJ cana-1458 93 15 open open ADJ cana-1458 93 16 ( ( PUNCT cana-1458 93 17 closed closed ADJ cana-1458 93 18 ) ) PUNCT cana-1458 93 19 set set VERB cana-1458 93 20 in in ADP cana-1458 93 21 ( ( PUNCT cana-1458 93 22 π‘ˆ π‘ˆ PROPN cana-1458 93 23 , , PUNCT cana-1458 93 24 οΏ½ οΏ½ PROPN cana-1458 93 25 Μƒ Μƒ PROPN cana-1458 93 26 οΏ½ οΏ½ PROPN cana-1458 93 27 ) ) PUNCT cana-1458 93 28 . . PUNCT cana-1458 94 1 proof proof NOUN cana-1458 94 2 . . PUNCT cana-1458 95 1 let let VERB cana-1458 95 2 { { PUNCT cana-1458 95 3 ( ( PUNCT cana-1458 95 4 𝐹 𝐹 PROPN cana-1458 95 5 , , PUNCT cana-1458 95 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 95 7 } } PUNCT cana-1458 95 8 be be AUX cana-1458 95 9 a a DET cana-1458 95 10 fuzzy fuzzy ADJ cana-1458 95 11 soft soft ADJ cana-1458 95 12 sequential sequential ADJ cana-1458 95 13 open open ADJ cana-1458 95 14 set set NOUN cana-1458 95 15 in in ADP cana-1458 95 16 fuzzy fuzzy ADJ cana-1458 95 17 soft soft ADJ cana-1458 95 18 sequential sequential ADJ cana-1458 95 19 topological topological ADJ cana-1458 95 20 space(π‘ˆ space(π‘ˆ PROPN cana-1458 95 21 , , PUNCT cana-1458 95 22 οΏ½ οΏ½ PROPN cana-1458 95 23 Μƒ Μƒ PROPN cana-1458 95 24 οΏ½ οΏ½ PROPN cana-1458 95 25 , , PUNCT cana-1458 95 26 𝐸 𝐸 PROPN cana-1458 95 27 ) ) PUNCT cana-1458 95 28 and and CCONJ cana-1458 95 29 ( ( PUNCT cana-1458 95 30 𝐹 𝐹 PROPN cana-1458 95 31 , , PUNCT cana-1458 95 32 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 95 33 be be AUX cana-1458 95 34 a a DET cana-1458 95 35 fuzzy fuzzy ADJ cana-1458 95 36 soft soft ADJ cana-1458 95 37 open open ADJ cana-1458 95 38 set set VERB cana-1458 95 39 in in ADP cana-1458 95 40 ( ( PUNCT cana-1458 95 41 π‘ˆ π‘ˆ PROPN cana-1458 95 42 , , PUNCT cana-1458 95 43 οΏ½ οΏ½ PROPN cana-1458 95 44 Μƒ Μƒ PROPN cana-1458 95 45 οΏ½ οΏ½ PROPN cana-1458 95 46 ) ) PUNCT cana-1458 95 47 . . PUNCT cana-1458 96 1 from from ADP cana-1458 96 2 the the DET cana-1458 96 3 proposition proposition NOUN cana-1458 96 4 3.12 3.12 NUM cana-1458 96 5 , , PUNCT cana-1458 96 6 fuzzy fuzzy ADJ cana-1458 96 7 soft soft ADJ cana-1458 96 8 topologies topology NOUN cana-1458 96 9 are be AUX cana-1458 96 10 induced induce VERB cana-1458 96 11 by by ADP cana-1458 96 12 fuzzy fuzzy ADJ cana-1458 96 13 soft soft ADJ cana-1458 96 14 sequential sequential ADJ cana-1458 96 15 topologies topology NOUN cana-1458 96 16 and and CCONJ cana-1458 96 17 any any DET cana-1458 96 18 fuzzy fuzzy ADJ cana-1458 96 19 soft soft ADJ cana-1458 96 20 topology topology NOUN cana-1458 96 21 οΏ½ οΏ½ PROPN cana-1458 96 22 Μƒ Μƒ PROPN cana-1458 96 23 οΏ½ οΏ½ PROPN cana-1458 96 24 on on ADP cana-1458 96 25 ( ( PUNCT cana-1458 96 26 π‘ˆ π‘ˆ PROPN cana-1458 96 27 , , PUNCT cana-1458 96 28 οΏ½ οΏ½ PROPN cana-1458 96 29 Μƒ Μƒ PROPN cana-1458 96 30 οΏ½ οΏ½ PROPN cana-1458 96 31 ) ) PUNCT cana-1458 96 32 can can AUX cana-1458 96 33 be be AUX cana-1458 96 34 considered consider VERB cana-1458 96 35 as as ADP cana-1458 96 36 a a DET cana-1458 96 37 component component NOUN cana-1458 96 38 of of ADP cana-1458 96 39 fuzzy fuzzy ADJ cana-1458 96 40 soft soft ADJ cana-1458 96 41 sequential sequential ADJ cana-1458 96 42 topology topology NOUN cana-1458 96 43 οΏ½ οΏ½ PROPN cana-1458 96 44 Μƒ Μƒ PROPN cana-1458 96 45 οΏ½ οΏ½ PROPN cana-1458 96 46 on on ADP cana-1458 96 47 π‘ˆ. π‘ˆ. PROPN cana-1458 96 48 hence hence ADV cana-1458 96 49 we we PRON cana-1458 96 50 get get VERB cana-1458 96 51 , , PUNCT cana-1458 96 52 for for ADP cana-1458 96 53 every every DET cana-1458 96 54 𝑛 𝑛 PROPN cana-1458 96 55 ∈ ∈ PROPN cana-1458 96 56 β„• β„• PROPN cana-1458 96 57 , , PUNCT cana-1458 96 58 ( ( PUNCT cana-1458 96 59 𝐹 𝐹 PROPN cana-1458 96 60 , , PUNCT cana-1458 96 61 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 96 62 is be AUX cana-1458 96 63 fuzzy fuzzy ADJ cana-1458 96 64 soft soft ADJ cana-1458 96 65 open open ADJ cana-1458 96 66 set set VERB cana-1458 96 67 in in ADP cana-1458 96 68 ( ( PUNCT cana-1458 96 69 π‘ˆ π‘ˆ PROPN cana-1458 96 70 , , PUNCT cana-1458 96 71 οΏ½ οΏ½ PROPN cana-1458 96 72 Μƒ Μƒ PROPN cana-1458 96 73 οΏ½ οΏ½ PROPN cana-1458 96 74 ) ) PUNCT cana-1458 96 75 . . PUNCT cana-1458 97 1 the the DET cana-1458 97 2 following follow VERB cana-1458 97 3 example example NOUN cana-1458 97 4 shows show VERB cana-1458 97 5 that that SCONJ cana-1458 97 6 , , PUNCT cana-1458 97 7 the the DET cana-1458 97 8 converse converse NOUN cana-1458 97 9 of of ADP cana-1458 97 10 the the DET cana-1458 97 11 above above ADJ cana-1458 97 12 proposition proposition NOUN cana-1458 97 13 need need AUX cana-1458 97 14 not not PART cana-1458 97 15 be be AUX cana-1458 97 16 true true ADJ cana-1458 97 17 . . PUNCT cana-1458 97 18 example example NOUN cana-1458 98 1 3.15 3.15 NUM cana-1458 98 2 . . PUNCT cana-1458 98 3 example example NOUN cana-1458 98 4 3.3 3.3 NUM cana-1458 98 5 shows show VERB cana-1458 98 6 that that SCONJ cana-1458 98 7 , , PUNCT cana-1458 98 8 { { PUNCT cana-1458 98 9 ( ( PUNCT cana-1458 98 10 𝐹 𝐹 PROPN cana-1458 98 11 , , PUNCT cana-1458 98 12 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 98 13 } } PUNCT cana-1458 98 14 is be AUX cana-1458 98 15 a a DET cana-1458 98 16 fuzzy fuzzy ADJ cana-1458 98 17 soft soft ADJ cana-1458 98 18 sequential sequential ADJ cana-1458 98 19 open open ADJ cana-1458 98 20 set set NOUN cana-1458 98 21 in in ADP cana-1458 98 22 ( ( PUNCT cana-1458 98 23 π‘ˆ π‘ˆ PROPN cana-1458 98 24 , , PUNCT cana-1458 98 25 οΏ½ οΏ½ PROPN cana-1458 98 26 Μƒ Μƒ PROPN cana-1458 98 27 οΏ½ οΏ½ PROPN cana-1458 98 28 , , PUNCT cana-1458 98 29 𝐸 𝐸 PROPN cana-1458 98 30 ) ) PUNCT cana-1458 98 31 implies imply VERB cana-1458 98 32 ( ( PUNCT cana-1458 98 33 𝐹 𝐹 PROPN cana-1458 98 34 , , PUNCT cana-1458 98 35 𝐴)1 𝐴)1 PROPN cana-1458 98 36 , , PUNCT cana-1458 98 37 ( ( PUNCT cana-1458 98 38 𝐹 𝐹 PROPN cana-1458 98 39 , , PUNCT cana-1458 98 40 𝐴)2 𝐴)2 PROPN cana-1458 98 41 , , PUNCT cana-1458 98 42 … … PUNCT cana-1458 98 43 .. .. PUNCT cana-1458 98 44 ( ( PUNCT cana-1458 98 45 𝐹 𝐹 PROPN cana-1458 98 46 , , PUNCT cana-1458 98 47 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 98 48 , , PUNCT cana-1458 98 49 … … PUNCT cana-1458 98 50 .. .. PUNCT cana-1458 98 51 is be AUX cana-1458 98 52 fuzzy fuzzy ADJ cana-1458 98 53 soft soft ADJ cana-1458 98 54 open open ADJ cana-1458 98 55 set set VERB cana-1458 98 56 in in ADP cana-1458 98 57 ( ( PUNCT cana-1458 98 58 π‘ˆ π‘ˆ PROPN cana-1458 98 59 , , PUNCT cana-1458 98 60 οΏ½ οΏ½ PROPN cana-1458 98 61 Μƒ Μƒ PROPN cana-1458 98 62 οΏ½ οΏ½ PROPN cana-1458 98 63 ) ) PUNCT cana-1458 98 64 . . PUNCT cana-1458 99 1 communications communication NOUN cana-1458 99 2 on on ADP cana-1458 99 3 applied apply VERB cana-1458 99 4 nonlinear nonlinear ADJ cana-1458 99 5 analysis analysis NOUN cana-1458 99 6 issn issn NOUN cana-1458 99 7 : : PUNCT cana-1458 99 8 1074 1074 NUM cana-1458 99 9 - - PUNCT cana-1458 99 10 133x 133x NUM cana-1458 99 11 vol vol NOUN cana-1458 99 12 31 31 NUM cana-1458 99 13 no no NOUN cana-1458 99 14 . . PUNCT cana-1458 100 1 8s 8s PROPN cana-1458 100 2 ( ( PUNCT cana-1458 100 3 2024 2024 NUM cana-1458 100 4 ) ) PUNCT cana-1458 100 5 91 91 NUM cana-1458 100 6 https://internationalpubls.com https://internationalpubls.com X cana-1458 100 7 now now ADV cana-1458 100 8 , , PUNCT cana-1458 100 9 let let VERB cana-1458 100 10 us we PRON cana-1458 100 11 take take VERB cana-1458 100 12 , , PUNCT cana-1458 100 13 for for ADP cana-1458 100 14 each each DET cana-1458 100 15 𝑛 𝑛 PRON cana-1458 100 16 ∈ ∈ PROPN cana-1458 100 17 β„• β„• PROPN cana-1458 100 18 , , PUNCT cana-1458 100 19 ( ( PUNCT cana-1458 100 20 𝐹 𝐹 PROPN cana-1458 100 21 , , PUNCT cana-1458 100 22 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 100 23 is be AUX cana-1458 100 24 fuzzy fuzzy ADJ cana-1458 100 25 soft soft ADJ cana-1458 100 26 open open ADJ cana-1458 100 27 set set VERB cana-1458 100 28 in in ADP cana-1458 100 29 ( ( PUNCT cana-1458 100 30 π‘ˆ π‘ˆ PROPN cana-1458 100 31 , , PUNCT cana-1458 100 32 οΏ½ οΏ½ PROPN cana-1458 100 33 Μƒ Μƒ PROPN cana-1458 100 34 οΏ½ οΏ½ PROPN cana-1458 100 35 ) ) PUNCT cana-1458 100 36 implies imply VERB cana-1458 100 37 { { PUNCT cana-1458 100 38 ( ( PUNCT cana-1458 100 39 𝐹 𝐹 PROPN cana-1458 100 40 , , PUNCT cana-1458 100 41 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 100 42 } } PUNCT cana-1458 100 43 is be AUX cana-1458 100 44 not not PART cana-1458 100 45 a a DET cana-1458 100 46 fuzzy fuzzy ADJ cana-1458 100 47 soft soft ADJ cana-1458 100 48 sequential sequential ADJ cana-1458 100 49 open open ADJ cana-1458 100 50 set set NOUN cana-1458 100 51 in in ADP cana-1458 100 52 ( ( PUNCT cana-1458 100 53 π‘ˆ π‘ˆ PROPN cana-1458 100 54 , , PUNCT cana-1458 100 55 οΏ½ οΏ½ PROPN cana-1458 100 56 Μƒ Μƒ PROPN cana-1458 100 57 οΏ½ οΏ½ PROPN cana-1458 100 58 , , PUNCT cana-1458 100 59 𝐸 𝐸 PROPN cana-1458 100 60 ) ) PUNCT cana-1458 100 61 . . PUNCT cana-1458 101 1 theorem theorem VERB cana-1458 101 2 3.16 3.16 NUM cana-1458 101 3 . . PUNCT cana-1458 102 1 arbitrary arbitrary ADJ cana-1458 102 2 intersection intersection NOUN cana-1458 102 3 of of ADP cana-1458 102 4 fuzzy fuzzy ADJ cana-1458 102 5 soft soft ADJ cana-1458 102 6 sequential sequential ADJ cana-1458 102 7 closed closed ADJ cana-1458 102 8 sets set NOUN cana-1458 102 9 is be AUX cana-1458 102 10 fuzzy fuzzy ADJ cana-1458 102 11 soft soft ADJ cana-1458 102 12 sequential sequential ADJ cana-1458 102 13 closed closed ADJ cana-1458 102 14 set set NOUN cana-1458 102 15 . . PUNCT cana-1458 103 1 proof proof NOUN cana-1458 103 2 . . PUNCT cana-1458 104 1 let let VERB cana-1458 104 2 { { PUNCT cana-1458 104 3 { { PUNCT cana-1458 104 4 ( ( PUNCT cana-1458 104 5 πΉπœ† πΉπœ† NOUN cana-1458 104 6 , , PUNCT cana-1458 104 7 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 104 8 } } PUNCT cana-1458 104 9 βˆ• βˆ• PUNCT cana-1458 104 10 πœ† πœ† PRON cana-1458 104 11 ∈ ∈ PROPN cana-1458 104 12 Ξ» Ξ» PROPN cana-1458 104 13 } } PUNCT cana-1458 104 14 be be VERB cana-1458 104 15 an an DET cana-1458 104 16 arbitrary arbitrary ADJ cana-1458 104 17 collection collection NOUN cana-1458 104 18 of of ADP cana-1458 104 19 fuzzy fuzzy ADJ cana-1458 104 20 soft soft ADJ cana-1458 104 21 sequential sequential ADJ cana-1458 104 22 closed closed ADJ cana-1458 104 23 sets set NOUN cana-1458 104 24 . . PUNCT cana-1458 105 1 we we PRON cana-1458 105 2 take take VERB cana-1458 105 3 { { PUNCT cana-1458 105 4 ( ( PUNCT cana-1458 105 5 𝐹 𝐹 PROPN cana-1458 105 6 , , PUNCT cana-1458 105 7 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 105 8 } } PUNCT cana-1458 105 9 = = SYM cana-1458 105 10 { { PUNCT cana-1458 105 11 ( ( PUNCT cana-1458 105 12 πΉπœ† πΉπœ† PROPN cana-1458 105 13 , , PUNCT cana-1458 105 14 𝐴)𝑛}πœ†βˆˆΞ» 𝐴)𝑛}πœ†βˆˆΞ» PROPN cana-1458 105 15 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 105 16 . . PUNCT cana-1458 106 1 then then ADV cana-1458 106 2 π‘ˆ π‘ˆ PROPN cana-1458 106 3 βˆ’ βˆ’ PROPN cana-1458 106 4 { { PUNCT cana-1458 106 5 ( ( PUNCT cana-1458 106 6 𝐹 𝐹 PROPN cana-1458 106 7 , , PUNCT cana-1458 106 8 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 106 9 } } PUNCT cana-1458 106 10 = = PUNCT cana-1458 106 11 π‘ˆ π‘ˆ PROPN cana-1458 106 12 βˆ’ βˆ’ PROPN cana-1458 106 13 { { PUNCT cana-1458 106 14 ( ( PUNCT cana-1458 106 15 πΉπœ† πΉπœ† NOUN cana-1458 106 16 , , PUNCT cana-1458 106 17 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 106 18 } } PUNCT cana-1458 107 1 = = NOUN cana-1458 107 2 πœ†βˆˆΞ» πœ†βˆˆΞ» NUM cana-1458 107 3 βˆ©Μƒ βˆ©Μƒ SYM cana-1458 107 4 ( ( PUNCT cana-1458 107 5 π‘ˆ π‘ˆ PROPN cana-1458 107 6 βˆ’ βˆ’ PROPN cana-1458 107 7 { { PUNCT cana-1458 107 8 ( ( PUNCT cana-1458 107 9 πΉπœ† πΉπœ† PROPN cana-1458 107 10 , , PUNCT cana-1458 107 11 𝐴)𝑛})πœ†βˆˆΞ» 𝐴)𝑛})πœ†βˆˆΞ» PROPN cana-1458 107 12 βˆͺΜƒ βˆͺΜƒ PROPN cana-1458 107 13 ( ( PUNCT cana-1458 107 14 by by ADP cana-1458 107 15 de de PROPN cana-1458 107 16 morgan morgan PROPN cana-1458 107 17 ’s ’s PART cana-1458 107 18 law).since law).since PROPN cana-1458 107 19 { { PUNCT cana-1458 107 20 ( ( PUNCT cana-1458 107 21 πΉπœ† πΉπœ† NOUN cana-1458 107 22 , , PUNCT cana-1458 107 23 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 107 24 } } PUNCT cana-1458 107 25 is be AUX cana-1458 107 26 fuzzy fuzzy ADJ cana-1458 107 27 soft soft ADJ cana-1458 107 28 sequential sequential ADJ cana-1458 107 29 closed close VERB cana-1458 107 30 set set NOUN cana-1458 107 31 in in ADP cana-1458 107 32 π‘ˆ π‘ˆ PROPN cana-1458 107 33 for for ADP cana-1458 107 34 all all PRON cana-1458 107 35 πœ† πœ† PRON cana-1458 107 36 ∈ ∈ PROPN cana-1458 107 37 Ξ» Ξ» PROPN cana-1458 107 38 , , PUNCT cana-1458 107 39 π‘ˆ π‘ˆ PROPN cana-1458 107 40 βˆ’ βˆ’ PROPN cana-1458 107 41 { { PUNCT cana-1458 107 42 ( ( PUNCT cana-1458 107 43 πΉπœ† πΉπœ† NOUN cana-1458 107 44 , , PUNCT cana-1458 107 45 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 107 46 } } PUNCT cana-1458 107 47 is be AUX cana-1458 107 48 fuzzy fuzzy ADJ cana-1458 107 49 soft soft ADJ cana-1458 107 50 sequential sequential ADJ cana-1458 107 51 open open ADJ cana-1458 107 52 set set NOUN cana-1458 107 53 in in ADP cana-1458 107 54 π‘ˆ. π‘ˆ. PROPN cana-1458 107 55 since since SCONJ cana-1458 107 56 arbitrary arbitrary ADJ cana-1458 107 57 union union NOUN cana-1458 107 58 of of ADP cana-1458 107 59 fuzzy fuzzy ADJ cana-1458 107 60 soft soft ADJ cana-1458 107 61 sequential sequential ADJ cana-1458 107 62 open open ADJ cana-1458 107 63 set set NOUN cana-1458 107 64 is be AUX cana-1458 107 65 fuzzy fuzzy ADJ cana-1458 107 66 soft soft ADJ cana-1458 107 67 sequential sequential ADJ cana-1458 107 68 open open NOUN cana-1458 107 69 , , PUNCT cana-1458 107 70 π‘ˆ π‘ˆ PROPN cana-1458 107 71 βˆ’ βˆ’ PROPN cana-1458 107 72 { { PUNCT cana-1458 107 73 ( ( PUNCT cana-1458 107 74 𝐹 𝐹 PROPN cana-1458 107 75 , , PUNCT cana-1458 107 76 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 107 77 } } PUNCT cana-1458 107 78 is be AUX cana-1458 107 79 fuzzy fuzzy ADJ cana-1458 107 80 soft soft ADJ cana-1458 107 81 sequential sequential ADJ cana-1458 107 82 open open ADJ cana-1458 107 83 set set NOUN cana-1458 107 84 . . PUNCT cana-1458 108 1 hence hence ADV cana-1458 108 2 { { PUNCT cana-1458 108 3 ( ( PUNCT cana-1458 108 4 𝐹 𝐹 PROPN cana-1458 108 5 , , PUNCT cana-1458 108 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 108 7 } } PUNCT cana-1458 108 8 is be AUX cana-1458 108 9 fuzzy fuzzy ADJ cana-1458 108 10 soft soft ADJ cana-1458 108 11 sequential sequential ADJ cana-1458 108 12 closed closed ADJ cana-1458 108 13 set set NOUN cana-1458 108 14 . . PUNCT cana-1458 109 1 theorem theorem VERB cana-1458 109 2 3.17 3.17 NUM cana-1458 109 3 . . PUNCT cana-1458 110 1 finite finite PROPN cana-1458 110 2 union union PROPN cana-1458 110 3 of of ADP cana-1458 110 4 fuzzy fuzzy ADJ cana-1458 110 5 soft soft ADJ cana-1458 110 6 sequential sequential ADJ cana-1458 110 7 closed closed ADJ cana-1458 110 8 sets set NOUN cana-1458 110 9 are be AUX cana-1458 110 10 fuzzy fuzzy ADJ cana-1458 110 11 soft soft ADJ cana-1458 110 12 sequential sequential ADJ cana-1458 110 13 closed closed ADJ cana-1458 110 14 . . PUNCT cana-1458 111 1 proof proof NOUN cana-1458 111 2 . . PUNCT cana-1458 112 1 let let VERB cana-1458 112 2 { { PUNCT cana-1458 112 3 ( ( PUNCT cana-1458 112 4 𝐹𝑖 𝐹𝑖 VERB cana-1458 112 5 , , PUNCT cana-1458 112 6 𝐴)𝑛}𝑖=1 𝐴)𝑛}𝑖=1 SCONJ cana-1458 112 7 π‘˜ π‘˜ PROPN cana-1458 112 8 be be AUX cana-1458 112 9 a a DET cana-1458 112 10 finite finite ADJ cana-1458 112 11 collection collection NOUN cana-1458 112 12 of of ADP cana-1458 112 13 fuzzy fuzzy ADJ cana-1458 112 14 soft soft ADJ cana-1458 112 15 sequential sequential ADJ cana-1458 112 16 closed closed ADJ cana-1458 112 17 sets set NOUN cana-1458 112 18 . . PUNCT cana-1458 113 1 let let VERB cana-1458 113 2 { { PUNCT cana-1458 113 3 ( ( PUNCT cana-1458 113 4 𝐹 𝐹 PROPN cana-1458 113 5 , , PUNCT cana-1458 113 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 113 7 } } PUNCT cana-1458 113 8 = = SYM cana-1458 113 9 βˆͺ̃𝑖=1 βˆͺ̃𝑖=1 PROPN cana-1458 113 10 π‘˜ π‘˜ X cana-1458 113 11 { { PUNCT cana-1458 113 12 ( ( PUNCT cana-1458 113 13 𝐹𝑖 𝐹𝑖 NOUN cana-1458 113 14 , , PUNCT cana-1458 113 15 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 113 16 } } PUNCT cana-1458 113 17 . . PUNCT cana-1458 114 1 then then ADV cana-1458 114 2 π‘ˆ π‘ˆ PROPN cana-1458 114 3 βˆ’ βˆ’ PROPN cana-1458 114 4 { { PUNCT cana-1458 114 5 ( ( PUNCT cana-1458 114 6 𝐹 𝐹 PROPN cana-1458 114 7 , , PUNCT cana-1458 114 8 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 114 9 } } PUNCT cana-1458 114 10 = = SYM cana-1458 114 11 π‘ˆ π‘ˆ NOUN cana-1458 114 12 βˆ’βˆͺ̃𝑖=1 βˆ’βˆͺ̃𝑖=1 VERB cana-1458 114 13 π‘˜ π‘˜ X cana-1458 114 14 { { PUNCT cana-1458 114 15 ( ( PUNCT cana-1458 114 16 𝐹𝑖 𝐹𝑖 NOUN cana-1458 114 17 , , PUNCT cana-1458 114 18 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 114 19 } } PUNCT cana-1458 114 20 = = NOUN cana-1458 114 21 βˆ©Μƒπ‘–=1 βˆ©Μƒπ‘–=1 X cana-1458 114 22 π‘˜ π‘˜ X cana-1458 114 23 ( ( PUNCT cana-1458 114 24 π‘ˆ π‘ˆ PROPN cana-1458 114 25 βˆ’ βˆ’ PROPN cana-1458 114 26 { { PUNCT cana-1458 114 27 ( ( PUNCT cana-1458 114 28 𝐹𝑖 𝐹𝑖 NOUN cana-1458 114 29 , , PUNCT cana-1458 114 30 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 114 31 } } PUNCT cana-1458 114 32 ( ( PUNCT cana-1458 114 33 by by ADP cana-1458 114 34 de de PROPN cana-1458 114 35 morgan morgan PROPN cana-1458 114 36 ’s ’s PART cana-1458 114 37 law law NOUN cana-1458 114 38 ) ) PUNCT cana-1458 114 39 . . PUNCT cana-1458 115 1 since since SCONJ cana-1458 115 2 { { PUNCT cana-1458 115 3 ( ( PUNCT cana-1458 115 4 𝐹𝑖 𝐹𝑖 NOUN cana-1458 115 5 , , PUNCT cana-1458 115 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 115 7 } } PUNCT cana-1458 115 8 is be AUX cana-1458 115 9 fuzzy fuzzy ADJ cana-1458 115 10 soft soft ADJ cana-1458 115 11 sequential sequential ADJ cana-1458 115 12 closed close VERB cana-1458 115 13 set set NOUN cana-1458 115 14 in in ADP cana-1458 115 15 π‘ˆ π‘ˆ PROPN cana-1458 115 16 , , PUNCT cana-1458 115 17 π‘ˆ π‘ˆ PROPN cana-1458 115 18 βˆ’ βˆ’ PROPN cana-1458 115 19 { { PUNCT cana-1458 115 20 ( ( PUNCT cana-1458 115 21 𝐹𝑖 𝐹𝑖 NOUN cana-1458 115 22 , , PUNCT cana-1458 115 23 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 115 24 } } PUNCT cana-1458 115 25 fuzzy fuzzy ADJ cana-1458 115 26 soft soft ADJ cana-1458 115 27 sequential sequential ADJ cana-1458 115 28 open open ADJ cana-1458 115 29 set set NOUN cana-1458 115 30 in in ADP cana-1458 115 31 π‘ˆ. π‘ˆ. PROPN cana-1458 115 32 since since SCONJ cana-1458 115 33 finite finite ADJ cana-1458 115 34 intersection intersection NOUN cana-1458 115 35 of of ADP cana-1458 115 36 fuzzy fuzzy ADJ cana-1458 115 37 soft soft ADJ cana-1458 115 38 sequential sequential ADJ cana-1458 115 39 open open ADJ cana-1458 115 40 set set NOUN cana-1458 115 41 is be AUX cana-1458 115 42 fuzzy fuzzy ADJ cana-1458 115 43 soft soft ADJ cana-1458 115 44 sequential sequential ADJ cana-1458 115 45 open open ADJ cana-1458 115 46 set set NOUN cana-1458 115 47 , , PUNCT cana-1458 115 48 π‘ˆ π‘ˆ PROPN cana-1458 115 49 βˆ’ βˆ’ PROPN cana-1458 115 50 { { PUNCT cana-1458 115 51 ( ( PUNCT cana-1458 115 52 𝐹 𝐹 PROPN cana-1458 115 53 , , PUNCT cana-1458 115 54 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 115 55 } } PUNCT cana-1458 115 56 is be AUX cana-1458 115 57 fuzzy fuzzy ADJ cana-1458 115 58 soft soft ADJ cana-1458 115 59 sequential sequential ADJ cana-1458 115 60 open open ADJ cana-1458 115 61 set set NOUN cana-1458 115 62 . . PUNCT cana-1458 116 1 hence hence ADV cana-1458 116 2 { { PUNCT cana-1458 116 3 ( ( PUNCT cana-1458 116 4 𝐹 𝐹 PROPN cana-1458 116 5 , , PUNCT cana-1458 116 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 116 7 } } PUNCT cana-1458 116 8 is be AUX cana-1458 116 9 fuzzy fuzzy ADJ cana-1458 116 10 soft soft ADJ cana-1458 116 11 sequential sequential ADJ cana-1458 116 12 closed close VERB cana-1458 116 13 set set NOUN cana-1458 116 14 . . PUNCT cana-1458 117 1 remark remark PROPN cana-1458 117 2 3.18 3.18 NUM cana-1458 117 3 . . PUNCT cana-1458 118 1 the the DET cana-1458 118 2 set set NOUN cana-1458 118 3 of of ADP cana-1458 118 4 all all DET cana-1458 118 5 fuzzy fuzzy ADJ cana-1458 118 6 soft soft ADJ cana-1458 118 7 sequential sequential ADJ cana-1458 118 8 closed closed ADJ cana-1458 118 9 sets set NOUN cana-1458 118 10 in in ADP cana-1458 118 11 fuzzy fuzzy ADJ cana-1458 118 12 soft soft ADJ cana-1458 118 13 sequential sequential ADJ cana-1458 118 14 topological topological ADJ cana-1458 118 15 space space NOUN cana-1458 118 16 ( ( PUNCT cana-1458 118 17 π‘ˆ π‘ˆ PROPN cana-1458 118 18 , , PUNCT cana-1458 118 19 οΏ½ οΏ½ PROPN cana-1458 118 20 Μƒ Μƒ PROPN cana-1458 118 21 οΏ½ οΏ½ PROPN cana-1458 118 22 , , PUNCT cana-1458 118 23 𝐸 𝐸 PROPN cana-1458 118 24 ) ) PUNCT cana-1458 118 25 , , PUNCT cana-1458 118 26 form form VERB cana-1458 118 27 a a DET cana-1458 118 28 fuzzy fuzzy ADJ cana-1458 118 29 soft soft ADJ cana-1458 118 30 sequential sequential ADJ cana-1458 118 31 topology topology NOUN cana-1458 118 32 on on ADP cana-1458 118 33 π‘ˆ. π‘ˆ. PROPN cana-1458 118 34 4 4 NUM cana-1458 118 35 . . PUNCT cana-1458 118 36 fuzzy fuzzy ADJ cana-1458 118 37 soft soft ADJ cana-1458 118 38 sequential sequential ADJ cana-1458 118 39 neighborhood neighborhood NOUN cana-1458 118 40 system system NOUN cana-1458 118 41 and and CCONJ cana-1458 118 42 properties property NOUN cana-1458 118 43 of of ADP cana-1458 118 44 fuzzy fuzzy ADJ cana-1458 118 45 soft soft ADJ cana-1458 118 46 sequential sequential ADJ cana-1458 118 47 quasicoincident quasicoincident NOUN cana-1458 118 48 of of ADP cana-1458 118 49 a a DET cana-1458 118 50 fuzzy fuzzy ADJ cana-1458 118 51 soft soft ADJ cana-1458 118 52 sequential sequential ADJ cana-1458 118 53 set set VERB cana-1458 118 54 definition definition NOUN cana-1458 118 55 4.1 4.1 NUM cana-1458 118 56 . . PUNCT cana-1458 119 1 let let AUX cana-1458 119 2 ( ( PUNCT cana-1458 119 3 π‘ˆ π‘ˆ PROPN cana-1458 119 4 , , PUNCT cana-1458 119 5 οΏ½ οΏ½ PROPN cana-1458 119 6 Μƒ Μƒ PROPN cana-1458 119 7 οΏ½ οΏ½ PROPN cana-1458 119 8 , , PUNCT cana-1458 119 9 𝐸 𝐸 PROPN cana-1458 119 10 ) ) PUNCT cana-1458 119 11 be be VERB cana-1458 119 12 a a DET cana-1458 119 13 fuzzy fuzzy ADJ cana-1458 119 14 soft soft ADJ cana-1458 119 15 sequential sequential ADJ cana-1458 119 16 topological topological ADJ cana-1458 119 17 space space NOUN cana-1458 119 18 over over ADP cana-1458 119 19 π‘ˆ π‘ˆ PROPN cana-1458 119 20 and and CCONJ cana-1458 119 21 { { PUNCT cana-1458 119 22 ( ( PUNCT cana-1458 119 23 𝐹 𝐹 PROPN cana-1458 119 24 , , PUNCT cana-1458 119 25 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 119 26 } } PUNCT cana-1458 119 27 be be AUX cana-1458 119 28 a a DET cana-1458 119 29 fuzzy fuzzy ADJ cana-1458 119 30 soft soft ADJ cana-1458 119 31 sequential sequential ADJ cana-1458 119 32 set set VERB cana-1458 119 33 over over ADP cana-1458 119 34 π‘ˆ. π‘ˆ. PROPN cana-1458 119 35 a a DET cana-1458 119 36 fuzzy fuzzy ADJ cana-1458 119 37 soft soft ADJ cana-1458 119 38 sequential sequential ADJ cana-1458 119 39 set set NOUN cana-1458 119 40 { { PUNCT cana-1458 119 41 ( ( PUNCT cana-1458 119 42 𝐹 𝐹 PROPN cana-1458 119 43 , , PUNCT cana-1458 119 44 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 119 45 } } PUNCT cana-1458 119 46 is be AUX cana-1458 119 47 called call VERB cana-1458 119 48 a a DET cana-1458 119 49 fuzzy fuzzy ADJ cana-1458 119 50 soft soft ADJ cana-1458 119 51 sequential sequential ADJ cana-1458 119 52 neighborhood neighborhood NOUN cana-1458 119 53 ( ( PUNCT cana-1458 119 54 briefly briefly ADV cana-1458 119 55 , , PUNCT cana-1458 119 56 𝐹𝑆𝑆-nbhd 𝐹𝑆𝑆-nbhd PROPN cana-1458 119 57 ) ) PUNCT cana-1458 119 58 of of ADP cana-1458 119 59 a a DET cana-1458 119 60 fuzzy fuzzy ADJ cana-1458 119 61 soft soft ADJ cana-1458 119 62 sequential sequential ADJ cana-1458 119 63 set set NOUN cana-1458 119 64 { { PUNCT cana-1458 119 65 ( ( PUNCT cana-1458 119 66 𝐺 𝐺 NOUN cana-1458 119 67 , , PUNCT cana-1458 119 68 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 119 69 } } PUNCT cana-1458 119 70 if if SCONJ cana-1458 119 71 and and CCONJ cana-1458 119 72 only only ADV cana-1458 119 73 if if SCONJ cana-1458 119 74 there there PRON cana-1458 119 75 exists exist VERB cana-1458 119 76 a a DET cana-1458 119 77 fuzzy fuzzy ADJ cana-1458 119 78 soft soft ADJ cana-1458 119 79 sequential sequential ADJ cana-1458 119 80 open open ADJ cana-1458 119 81 set set NOUN cana-1458 119 82 { { PUNCT cana-1458 119 83 ( ( PUNCT cana-1458 119 84 𝐻 𝐻 PROPN cana-1458 119 85 , , PUNCT cana-1458 119 86 𝐢)𝑛 𝐢)𝑛 NOUN cana-1458 119 87 } } PUNCT cana-1458 119 88 such such ADJ cana-1458 119 89 that that SCONJ cana-1458 119 90 { { PUNCT cana-1458 119 91 ( ( PUNCT cana-1458 119 92 𝐺 𝐺 NOUN cana-1458 119 93 , , PUNCT cana-1458 119 94 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 119 95 } } PUNCT cana-1458 119 96 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 119 97 { { PUNCT cana-1458 119 98 ( ( PUNCT cana-1458 119 99 𝐻 𝐻 PROPN cana-1458 119 100 , , PUNCT cana-1458 119 101 𝐢)𝑛 𝐢)𝑛 NUM cana-1458 119 102 } } PUNCT cana-1458 119 103 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 119 104 { { PUNCT cana-1458 119 105 ( ( PUNCT cana-1458 119 106 𝐹 𝐹 PROPN cana-1458 119 107 , , PUNCT cana-1458 119 108 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 119 109 } } PUNCT cana-1458 119 110 . . PUNCT cana-1458 120 1 a a DET cana-1458 120 2 fuzzy fuzzy ADJ cana-1458 120 3 soft soft ADJ cana-1458 120 4 sequential sequential ADJ cana-1458 120 5 neighborhood neighborhood NOUN cana-1458 120 6 { { PUNCT cana-1458 120 7 ( ( PUNCT cana-1458 120 8 𝐹 𝐹 PROPN cana-1458 120 9 , , PUNCT cana-1458 120 10 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 120 11 } } PUNCT cana-1458 120 12 is be AUX cana-1458 120 13 said say VERB cana-1458 120 14 to to PART cana-1458 120 15 be be AUX cana-1458 120 16 an an DET cana-1458 120 17 open open ADJ cana-1458 120 18 if if SCONJ cana-1458 120 19 and and CCONJ cana-1458 120 20 only only ADV cana-1458 120 21 if if SCONJ cana-1458 120 22 { { PUNCT cana-1458 120 23 ( ( PUNCT cana-1458 120 24 𝐹 𝐹 PROPN cana-1458 120 25 , , PUNCT cana-1458 120 26 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 120 27 } } PUNCT cana-1458 120 28 is be AUX cana-1458 120 29 fuzzy fuzzy ADJ cana-1458 120 30 soft soft ADJ cana-1458 120 31 sequential sequential ADJ cana-1458 120 32 open open ADJ cana-1458 120 33 set set NOUN cana-1458 120 34 . . PUNCT cana-1458 121 1 the the DET cana-1458 121 2 collection collection NOUN cana-1458 121 3 of of ADP cana-1458 121 4 all all DET cana-1458 121 5 fuzzy fuzzy ADJ cana-1458 121 6 soft soft ADJ cana-1458 121 7 sequential sequential ADJ cana-1458 121 8 neighborhood neighborhood NOUN cana-1458 121 9 of of ADP cana-1458 121 10 { { PUNCT cana-1458 121 11 ( ( PUNCT cana-1458 121 12 𝐹 𝐹 PROPN cana-1458 121 13 , , PUNCT cana-1458 121 14 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 121 15 } } PUNCT cana-1458 121 16 is be AUX cana-1458 121 17 called call VERB cana-1458 121 18 the the DET cana-1458 121 19 fuzzy fuzzy ADJ cana-1458 121 20 soft soft ADJ cana-1458 121 21 sequential sequential ADJ cana-1458 121 22 neighborhood neighborhood NOUN cana-1458 121 23 system system NOUN cana-1458 121 24 of of ADP cana-1458 121 25 { { PUNCT cana-1458 121 26 ( ( PUNCT cana-1458 121 27 𝐹 𝐹 PROPN cana-1458 121 28 , , PUNCT cana-1458 121 29 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 121 30 } } PUNCT cana-1458 121 31 up up ADP cana-1458 121 32 to to ADP cana-1458 121 33 topology topology NOUN cana-1458 121 34 οΏ½ οΏ½ PROPN cana-1458 121 35 Μƒ Μƒ PROPN cana-1458 121 36 οΏ½ οΏ½ PROPN cana-1458 121 37 and and CCONJ cana-1458 121 38 is be AUX cana-1458 121 39 denoted denote VERB cana-1458 121 40 by by ADP cana-1458 121 41 𝒩 𝒩 PROPN cana-1458 121 42 { { PUNCT cana-1458 121 43 ( ( PUNCT cana-1458 121 44 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 ADJ cana-1458 121 45 } } PUNCT cana-1458 121 46 . . PUNCT cana-1458 122 1 theorem theorem VERB cana-1458 122 2 4.2 4.2 NUM cana-1458 122 3 . . PUNCT cana-1458 123 1 a a DET cana-1458 123 2 fuzzy fuzzy ADJ cana-1458 123 3 soft soft ADJ cana-1458 123 4 sequential sequential ADJ cana-1458 123 5 set set NOUN cana-1458 123 6 { { PUNCT cana-1458 123 7 ( ( PUNCT cana-1458 123 8 𝐹 𝐹 PROPN cana-1458 123 9 , , PUNCT cana-1458 123 10 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 123 11 } } PUNCT cana-1458 123 12 over over ADP cana-1458 123 13 π‘ˆ π‘ˆ PROPN cana-1458 123 14 is be AUX cana-1458 123 15 a a DET cana-1458 123 16 fuzzy fuzzy ADJ cana-1458 123 17 soft soft ADJ cana-1458 123 18 sequential sequential ADJ cana-1458 123 19 open open ADJ cana-1458 123 20 set set NOUN cana-1458 123 21 if if SCONJ cana-1458 123 22 and and CCONJ cana-1458 123 23 only only ADV cana-1458 123 24 if if SCONJ cana-1458 123 25 { { PUNCT cana-1458 123 26 ( ( PUNCT cana-1458 123 27 𝐹 𝐹 PROPN cana-1458 123 28 , , PUNCT cana-1458 123 29 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 123 30 } } PUNCT cana-1458 123 31 is be AUX cana-1458 123 32 a a DET cana-1458 123 33 fuzzy fuzzy ADJ cana-1458 123 34 soft soft ADJ cana-1458 123 35 sequential sequential ADJ cana-1458 123 36 neighborhood neighborhood NOUN cana-1458 123 37 of of ADP cana-1458 123 38 each each DET cana-1458 123 39 fuzzy fuzzy ADJ cana-1458 123 40 soft soft ADJ cana-1458 123 41 sequential sequential ADJ cana-1458 123 42 set set NOUN cana-1458 123 43 { { PUNCT cana-1458 123 44 ( ( PUNCT cana-1458 123 45 𝐺 𝐺 NOUN cana-1458 123 46 , , PUNCT cana-1458 123 47 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 123 48 } } PUNCT cana-1458 123 49 contained contain VERB cana-1458 123 50 in in ADP cana-1458 123 51 { { PUNCT cana-1458 123 52 ( ( PUNCT cana-1458 123 53 𝐹 𝐹 PROPN cana-1458 123 54 , , PUNCT cana-1458 123 55 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 123 56 } } PUNCT cana-1458 123 57 . . PUNCT cana-1458 124 1 proof proof NOUN cana-1458 124 2 . . PUNCT cana-1458 125 1 necessary necessary ADJ cana-1458 125 2 part part NOUN cana-1458 125 3 . . PUNCT cana-1458 126 1 let let AUX cana-1458 126 2 { { PUNCT cana-1458 126 3 ( ( PUNCT cana-1458 126 4 𝐹 𝐹 PROPN cana-1458 126 5 , , PUNCT cana-1458 126 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 126 7 } } PUNCT cana-1458 126 8 be be AUX cana-1458 126 9 a a DET cana-1458 126 10 fuzzy fuzzy ADJ cana-1458 126 11 soft soft ADJ cana-1458 126 12 sequential sequential ADJ cana-1458 126 13 open open ADJ cana-1458 126 14 set set NOUN cana-1458 126 15 and and CCONJ cana-1458 126 16 { { PUNCT cana-1458 126 17 ( ( PUNCT cana-1458 126 18 𝐺 𝐺 NOUN cana-1458 126 19 , , PUNCT cana-1458 126 20 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 126 21 } } PUNCT cana-1458 126 22 be be VERB cana-1458 126 23 any any DET cana-1458 126 24 fuzzy fuzzy ADJ cana-1458 126 25 soft soft ADJ cana-1458 126 26 sequential sequential ADJ cana-1458 126 27 set set NOUN cana-1458 126 28 contained contain VERB cana-1458 126 29 in in ADP cana-1458 126 30 { { PUNCT cana-1458 126 31 ( ( PUNCT cana-1458 126 32 𝐹 𝐹 PROPN cana-1458 126 33 , , PUNCT cana-1458 126 34 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 126 35 } } PUNCT cana-1458 126 36 . . PUNCT cana-1458 127 1 since since SCONJ cana-1458 127 2 { { PUNCT cana-1458 127 3 ( ( PUNCT cana-1458 127 4 𝐺 𝐺 NOUN cana-1458 127 5 , , PUNCT cana-1458 127 6 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 127 7 } } PUNCT cana-1458 127 8 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 127 9 { { PUNCT cana-1458 127 10 ( ( PUNCT cana-1458 127 11 𝐹 𝐹 PROPN cana-1458 127 12 , , PUNCT cana-1458 127 13 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 127 14 } } PUNCT cana-1458 127 15 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 127 16 { { PUNCT cana-1458 127 17 ( ( PUNCT cana-1458 127 18 𝐹 𝐹 PROPN cana-1458 127 19 , , PUNCT cana-1458 127 20 𝐴)𝑛 𝐴)𝑛 PROPN cana-1458 127 21 } } PUNCT cana-1458 127 22 , , PUNCT cana-1458 127 23 { { PUNCT cana-1458 127 24 ( ( PUNCT cana-1458 127 25 𝐹 𝐹 PROPN cana-1458 127 26 , , PUNCT cana-1458 127 27 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 127 28 } } PUNCT cana-1458 127 29 is be AUX cana-1458 127 30 a a DET cana-1458 127 31 fuzzy fuzzy ADJ cana-1458 127 32 soft soft ADJ cana-1458 127 33 sequential sequential ADJ cana-1458 127 34 neighborhood neighborhood NOUN cana-1458 127 35 of of ADP cana-1458 127 36 each each PRON cana-1458 127 37 { { PUNCT cana-1458 127 38 ( ( PUNCT cana-1458 127 39 𝐺 𝐺 NOUN cana-1458 127 40 , , PUNCT cana-1458 127 41 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 127 42 } } PUNCT cana-1458 127 43 contained contain VERB cana-1458 127 44 in in ADP cana-1458 127 45 { { PUNCT cana-1458 127 46 ( ( PUNCT cana-1458 127 47 𝐹 𝐹 PROPN cana-1458 127 48 , , PUNCT cana-1458 127 49 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 127 50 } } PUNCT cana-1458 127 51 . . PUNCT cana-1458 128 1 communications communication NOUN cana-1458 128 2 on on ADP cana-1458 128 3 applied apply VERB cana-1458 128 4 nonlinear nonlinear ADJ cana-1458 128 5 analysis analysis NOUN cana-1458 128 6 issn issn NOUN cana-1458 128 7 : : PUNCT cana-1458 128 8 1074 1074 NUM cana-1458 128 9 - - PUNCT cana-1458 128 10 133x 133x NUM cana-1458 128 11 vol vol NOUN cana-1458 128 12 31 31 NUM cana-1458 128 13 no no NOUN cana-1458 128 14 . . PUNCT cana-1458 129 1 8s 8s PROPN cana-1458 129 2 ( ( PUNCT cana-1458 129 3 2024 2024 NUM cana-1458 129 4 ) ) PUNCT cana-1458 129 5 92 92 NUM cana-1458 129 6 https://internationalpubls.com https://internationalpubls.com NUM cana-1458 129 7 sufficient sufficient ADJ cana-1458 129 8 part part NOUN cana-1458 129 9 . . PUNCT cana-1458 130 1 let let VERB cana-1458 130 2 { { PUNCT cana-1458 130 3 ( ( PUNCT cana-1458 130 4 𝐹 𝐹 PROPN cana-1458 130 5 , , PUNCT cana-1458 130 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 130 7 } } PUNCT cana-1458 130 8 be be AUX cana-1458 130 9 a a DET cana-1458 130 10 fuzzy fuzzy ADJ cana-1458 130 11 soft soft ADJ cana-1458 130 12 sequential sequential ADJ cana-1458 130 13 neighborhood neighborhood NOUN cana-1458 130 14 for for ADP cana-1458 130 15 every every DET cana-1458 130 16 fuzzy fuzzy ADJ cana-1458 130 17 soft soft ADJ cana-1458 130 18 sequential sequential ADJ cana-1458 130 19 set set NOUN cana-1458 130 20 contained contain VERB cana-1458 130 21 it it PRON cana-1458 130 22 . . PUNCT cana-1458 131 1 since since SCONJ cana-1458 131 2 { { PUNCT cana-1458 131 3 ( ( PUNCT cana-1458 131 4 𝐹 𝐹 PROPN cana-1458 131 5 , , PUNCT cana-1458 131 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 131 7 } } PUNCT cana-1458 131 8 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 131 9 { { PUNCT cana-1458 131 10 ( ( PUNCT cana-1458 131 11 𝐹 𝐹 PROPN cana-1458 131 12 , , PUNCT cana-1458 131 13 𝐴)𝑛 𝐴)𝑛 PROPN cana-1458 131 14 } } PUNCT cana-1458 131 15 , , PUNCT cana-1458 131 16 there there PRON cana-1458 131 17 exists exist VERB cana-1458 131 18 a a DET cana-1458 131 19 fuzzy fuzzy ADJ cana-1458 131 20 soft soft ADJ cana-1458 131 21 sequential sequential ADJ cana-1458 131 22 open open ADJ cana-1458 131 23 set set NOUN cana-1458 131 24 { { PUNCT cana-1458 131 25 ( ( PUNCT cana-1458 131 26 𝐺 𝐺 NOUN cana-1458 131 27 , , PUNCT cana-1458 131 28 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 131 29 } } PUNCT cana-1458 131 30 such such ADJ cana-1458 131 31 that that SCONJ cana-1458 131 32 { { PUNCT cana-1458 131 33 ( ( PUNCT cana-1458 131 34 𝐹 𝐹 PROPN cana-1458 131 35 , , PUNCT cana-1458 131 36 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 131 37 } } PUNCT cana-1458 131 38 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 131 39 { { PUNCT cana-1458 131 40 ( ( PUNCT cana-1458 131 41 𝐺 𝐺 NOUN cana-1458 131 42 , , PUNCT cana-1458 131 43 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 131 44 } } PUNCT cana-1458 131 45 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 131 46 { { PUNCT cana-1458 131 47 ( ( PUNCT cana-1458 131 48 𝐹 𝐹 PROPN cana-1458 131 49 , , PUNCT cana-1458 131 50 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 131 51 } } PUNCT cana-1458 131 52 .hence .hence ADP cana-1458 131 53 { { PUNCT cana-1458 131 54 ( ( PUNCT cana-1458 131 55 𝐹 𝐹 PROPN cana-1458 131 56 , , PUNCT cana-1458 131 57 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 131 58 } } PUNCT cana-1458 131 59 = = SYM cana-1458 131 60 { { PUNCT cana-1458 131 61 ( ( PUNCT cana-1458 131 62 𝐺 𝐺 NOUN cana-1458 131 63 , , PUNCT cana-1458 131 64 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 131 65 } } PUNCT cana-1458 131 66 and and CCONJ cana-1458 131 67 { { PUNCT cana-1458 131 68 ( ( PUNCT cana-1458 131 69 𝐹 𝐹 PROPN cana-1458 131 70 , , PUNCT cana-1458 131 71 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 131 72 } } PUNCT cana-1458 131 73 is be AUX cana-1458 131 74 fuzzy fuzzy ADJ cana-1458 131 75 soft soft ADJ cana-1458 131 76 sequential sequential ADJ cana-1458 131 77 open open ADJ cana-1458 131 78 set set NOUN cana-1458 131 79 . . PUNCT cana-1458 132 1 theorem theorem VERB cana-1458 132 2 4.3 4.3 NUM cana-1458 132 3 if if SCONJ cana-1458 132 4 𝒩 𝒩 PROPN cana-1458 132 5 { { PUNCT cana-1458 132 6 ( ( PUNCT cana-1458 132 7 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 ADJ cana-1458 132 8 } } PUNCT cana-1458 132 9 is be AUX cana-1458 132 10 the the DET cana-1458 132 11 fuzzy fuzzy ADJ cana-1458 132 12 soft soft ADJ cana-1458 132 13 sequential sequential ADJ cana-1458 132 14 neighborhood neighborhood NOUN cana-1458 132 15 system system NOUN cana-1458 132 16 of of ADP cana-1458 132 17 fuzzy fuzzy ADJ cana-1458 132 18 soft soft ADJ cana-1458 132 19 sequential sequential ADJ cana-1458 132 20 set set NOUN cana-1458 132 21 { { PUNCT cana-1458 132 22 ( ( PUNCT cana-1458 132 23 𝐹 𝐹 PROPN cana-1458 132 24 , , PUNCT cana-1458 132 25 𝐴)𝑛 𝐴)𝑛 PROPN cana-1458 132 26 } } PUNCT cana-1458 132 27 , , PUNCT cana-1458 132 28 then then ADV cana-1458 132 29 ( ( PUNCT cana-1458 132 30 i i NOUN cana-1458 132 31 ) ) PUNCT cana-1458 132 32 finite finite VERB cana-1458 132 33 intersection intersection NOUN cana-1458 132 34 of of ADP cana-1458 132 35 members member NOUN cana-1458 132 36 of of ADP cana-1458 132 37 𝒩 𝒩 PROPN cana-1458 132 38 { { PUNCT cana-1458 132 39 ( ( PUNCT cana-1458 132 40 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 ADJ cana-1458 132 41 } } PUNCT cana-1458 132 42 belongs belong VERB cana-1458 132 43 to to ADP cana-1458 132 44 𝒩 𝒩 PROPN cana-1458 132 45 { { PUNCT cana-1458 132 46 ( ( PUNCT cana-1458 132 47 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 NOUN cana-1458 132 48 } } PUNCT cana-1458 132 49 . . PUNCT cana-1458 133 1 ( ( PUNCT cana-1458 133 2 ii ii NOUN cana-1458 133 3 ) ) PUNCT cana-1458 133 4 each each DET cana-1458 133 5 fuzzy fuzzy ADJ cana-1458 133 6 soft soft ADJ cana-1458 133 7 sequential sequential ADJ cana-1458 133 8 set set NOUN cana-1458 133 9 which which PRON cana-1458 133 10 contains contain VERB cana-1458 133 11 a a DET cana-1458 133 12 member member NOUN cana-1458 133 13 of of ADP cana-1458 133 14 𝒩 𝒩 PROPN cana-1458 133 15 { { PUNCT cana-1458 133 16 ( ( PUNCT cana-1458 133 17 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 ADJ cana-1458 133 18 } } PUNCT cana-1458 133 19 belongs belong VERB cana-1458 133 20 to to ADP cana-1458 133 21 𝒩 𝒩 PROPN cana-1458 133 22 { { PUNCT cana-1458 133 23 ( ( PUNCT cana-1458 133 24 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 ADJ cana-1458 133 25 } } PUNCT cana-1458 133 26 . . PUNCT cana-1458 134 1 proof proof NOUN cana-1458 134 2 . . PUNCT cana-1458 135 1 ( ( PUNCT cana-1458 135 2 i i NOUN cana-1458 135 3 ) ) PUNCT cana-1458 135 4 if if SCONJ cana-1458 135 5 { { PUNCT cana-1458 135 6 ( ( PUNCT cana-1458 135 7 𝐺 𝐺 NOUN cana-1458 135 8 , , PUNCT cana-1458 135 9 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 135 10 } } PUNCT cana-1458 135 11 , , PUNCT cana-1458 135 12 { { PUNCT cana-1458 135 13 ( ( PUNCT cana-1458 135 14 𝐻 𝐻 PROPN cana-1458 135 15 , , PUNCT cana-1458 135 16 𝐢)𝑛 𝐢)𝑛 NUM cana-1458 135 17 } } PUNCT cana-1458 135 18 ∈ ∈ PROPN cana-1458 135 19 𝒩 𝒩 PROPN cana-1458 135 20 { { PUNCT cana-1458 135 21 ( ( PUNCT cana-1458 135 22 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 NOUN cana-1458 135 23 } } PUNCT cana-1458 135 24 . . PUNCT cana-1458 136 1 then then ADV cana-1458 136 2 there there PRON cana-1458 136 3 exists exist VERB cana-1458 136 4 { { PUNCT cana-1458 136 5 ( ( PUNCT cana-1458 136 6 𝐺′ 𝐺′ PROPN cana-1458 136 7 , , PUNCT cana-1458 136 8 𝐡′)𝑛 𝐡′)𝑛 PROPN cana-1458 136 9 } } PUNCT cana-1458 136 10 , , PUNCT cana-1458 136 11 { { PUNCT cana-1458 136 12 ( ( PUNCT cana-1458 136 13 𝐻′ 𝐻′ ADV cana-1458 136 14 , , PUNCT cana-1458 136 15 𝐢′)𝑛 𝐢′)𝑛 NOUN cana-1458 136 16 } } PUNCT cana-1458 136 17 ∈ ∈ PROPN cana-1458 136 18 οΏ½ οΏ½ PROPN cana-1458 136 19 Μƒ Μƒ NOUN cana-1458 136 20 οΏ½ οΏ½ PROPN cana-1458 136 21 such such ADJ cana-1458 136 22 that that SCONJ cana-1458 136 23 { { PUNCT cana-1458 136 24 ( ( PUNCT cana-1458 136 25 𝐹 𝐹 PROPN cana-1458 136 26 , , PUNCT cana-1458 136 27 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 136 28 } } PUNCT cana-1458 136 29 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 136 30 { { PUNCT cana-1458 136 31 ( ( PUNCT cana-1458 136 32 𝐺′ 𝐺′ PROPN cana-1458 136 33 , , PUNCT cana-1458 136 34 𝐡′)𝑛 𝐡′)𝑛 PROPN cana-1458 136 35 } } PUNCT cana-1458 136 36 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 136 37 { { PUNCT cana-1458 136 38 ( ( PUNCT cana-1458 136 39 𝐺 𝐺 NOUN cana-1458 136 40 , , PUNCT cana-1458 136 41 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 136 42 } } PUNCT cana-1458 136 43 and and CCONJ cana-1458 136 44 { { PUNCT cana-1458 136 45 ( ( PUNCT cana-1458 136 46 𝐹 𝐹 PROPN cana-1458 136 47 , , PUNCT cana-1458 136 48 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 136 49 } } PUNCT cana-1458 136 50 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 136 51 { { PUNCT cana-1458 136 52 ( ( PUNCT cana-1458 136 53 𝐻′ 𝐻′ ADV cana-1458 136 54 , , PUNCT cana-1458 136 55 𝐢′)𝑛 𝐢′)𝑛 NOUN cana-1458 136 56 } } PUNCT cana-1458 136 57 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 136 58 { { PUNCT cana-1458 136 59 ( ( PUNCT cana-1458 136 60 𝐻 𝐻 PROPN cana-1458 136 61 , , PUNCT cana-1458 136 62 𝐢)𝑛 𝐢)𝑛 NUM cana-1458 136 63 } } PUNCT cana-1458 136 64 . . PUNCT cana-1458 137 1 since since SCONJ cana-1458 137 2 { { PUNCT cana-1458 137 3 ( ( PUNCT cana-1458 137 4 𝐺′ 𝐺′ PROPN cana-1458 137 5 , , PUNCT cana-1458 137 6 𝐡′)𝑛 𝐡′)𝑛 PROPN cana-1458 137 7 } } PUNCT cana-1458 137 8 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 137 9 { { PUNCT cana-1458 137 10 ( ( PUNCT cana-1458 137 11 𝐻′ 𝐻′ ADV cana-1458 137 12 , , PUNCT cana-1458 137 13 𝐢′)𝑛 𝐢′)𝑛 NOUN cana-1458 137 14 } } PUNCT cana-1458 137 15 ∈ ∈ PROPN cana-1458 137 16 οΏ½ οΏ½ PROPN cana-1458 137 17 Μƒ Μƒ PROPN cana-1458 137 18 οΏ½ οΏ½ PROPN cana-1458 137 19 , , PUNCT cana-1458 137 20 { { PUNCT cana-1458 137 21 ( ( PUNCT cana-1458 137 22 𝐹 𝐹 PROPN cana-1458 137 23 , , PUNCT cana-1458 137 24 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 137 25 } } PUNCT cana-1458 137 26 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 137 27 { { PUNCT cana-1458 137 28 ( ( PUNCT cana-1458 137 29 𝐺′ 𝐺′ PROPN cana-1458 137 30 , , PUNCT cana-1458 137 31 𝐡′)𝑛 𝐡′)𝑛 PROPN cana-1458 137 32 } } PUNCT cana-1458 137 33 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 137 34 { { PUNCT cana-1458 137 35 ( ( PUNCT cana-1458 137 36 𝐻′ 𝐻′ ADV cana-1458 137 37 , , PUNCT cana-1458 137 38 𝐢′)𝑛 𝐢′)𝑛 NOUN cana-1458 137 39 } } PUNCT cana-1458 137 40 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 137 41 { { PUNCT cana-1458 137 42 ( ( PUNCT cana-1458 137 43 𝐺 𝐺 NOUN cana-1458 137 44 , , PUNCT cana-1458 137 45 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 137 46 } } PUNCT cana-1458 137 47 βˆ©Μƒ βˆ©Μƒ SYM cana-1458 137 48 { { PUNCT cana-1458 137 49 ( ( PUNCT cana-1458 137 50 𝐻 𝐻 PROPN cana-1458 137 51 , , PUNCT cana-1458 137 52 𝐢)𝑛 𝐢)𝑛 NUM cana-1458 137 53 } } PUNCT cana-1458 137 54 . . PUNCT cana-1458 138 1 hence hence ADV cana-1458 138 2 { { PUNCT cana-1458 138 3 ( ( PUNCT cana-1458 138 4 𝐺 𝐺 NOUN cana-1458 138 5 , , PUNCT cana-1458 138 6 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 138 7 } } PUNCT cana-1458 138 8 βˆ©Μƒ βˆ©Μƒ SYM cana-1458 138 9 { { PUNCT cana-1458 138 10 ( ( PUNCT cana-1458 138 11 𝐻 𝐻 PROPN cana-1458 138 12 , , PUNCT cana-1458 138 13 𝐢)𝑛 𝐢)𝑛 NUM cana-1458 138 14 } } PUNCT cana-1458 138 15 ∈ ∈ PROPN cana-1458 138 16 𝒩 𝒩 PROPN cana-1458 138 17 { { PUNCT cana-1458 138 18 ( ( PUNCT cana-1458 138 19 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 NOUN cana-1458 138 20 } } PUNCT cana-1458 138 21 . . PUNCT cana-1458 139 1 ( ( PUNCT cana-1458 139 2 ii ii NOUN cana-1458 139 3 ) ) PUNCT cana-1458 139 4 let let AUX cana-1458 139 5 { { PUNCT cana-1458 139 6 ( ( PUNCT cana-1458 139 7 𝐺 𝐺 NOUN cana-1458 139 8 , , PUNCT cana-1458 139 9 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 139 10 } } PUNCT cana-1458 139 11 ∈ ∈ PROPN cana-1458 139 12 𝒩 𝒩 PROPN cana-1458 139 13 { { PUNCT cana-1458 139 14 ( ( PUNCT cana-1458 139 15 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 NOUN cana-1458 139 16 } } PUNCT cana-1458 139 17 and and CCONJ cana-1458 139 18 { { PUNCT cana-1458 139 19 ( ( PUNCT cana-1458 139 20 𝐻 𝐻 PROPN cana-1458 139 21 , , PUNCT cana-1458 139 22 𝐢)𝑛 𝐢)𝑛 PUNCT cana-1458 139 23 } } PUNCT cana-1458 139 24 be be AUX cana-1458 139 25 a a DET cana-1458 139 26 fuzzy fuzzy ADJ cana-1458 139 27 soft soft ADJ cana-1458 139 28 sequential sequential ADJ cana-1458 139 29 set set NOUN cana-1458 139 30 which which PRON cana-1458 139 31 contains contain VERB cana-1458 139 32 { { PUNCT cana-1458 139 33 ( ( PUNCT cana-1458 139 34 𝐺 𝐺 NOUN cana-1458 139 35 , , PUNCT cana-1458 139 36 𝐡)𝑛 𝐡)𝑛 NUM cana-1458 139 37 } } PUNCT cana-1458 139 38 . . PUNCT cana-1458 140 1 since since SCONJ cana-1458 140 2 { { PUNCT cana-1458 140 3 ( ( PUNCT cana-1458 140 4 𝐺 𝐺 NOUN cana-1458 140 5 , , PUNCT cana-1458 140 6 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 140 7 } } PUNCT cana-1458 140 8 ∈ ∈ PROPN cana-1458 140 9 𝒩 𝒩 PROPN cana-1458 140 10 { { PUNCT cana-1458 140 11 ( ( PUNCT cana-1458 140 12 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 NOUN cana-1458 140 13 } } PUNCT cana-1458 140 14 , , PUNCT cana-1458 140 15 there there PRON cana-1458 140 16 exists exist VERB cana-1458 140 17 a a DET cana-1458 140 18 fuzzy fuzzy ADJ cana-1458 140 19 soft soft ADJ cana-1458 140 20 sequential sequential ADJ cana-1458 140 21 open open ADJ cana-1458 140 22 set set NOUN cana-1458 140 23 { { PUNCT cana-1458 140 24 ( ( PUNCT cana-1458 140 25 𝐺′ 𝐺′ PROPN cana-1458 140 26 , , PUNCT cana-1458 140 27 𝐡′)𝑛 𝐡′)𝑛 PROPN cana-1458 140 28 } } PUNCT cana-1458 140 29 such such ADJ cana-1458 140 30 that that SCONJ cana-1458 140 31 { { PUNCT cana-1458 140 32 ( ( PUNCT cana-1458 140 33 𝐹 𝐹 PROPN cana-1458 140 34 , , PUNCT cana-1458 140 35 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 140 36 } } PUNCT cana-1458 140 37 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 140 38 { { PUNCT cana-1458 140 39 ( ( PUNCT cana-1458 140 40 𝐺′ 𝐺′ PROPN cana-1458 140 41 , , PUNCT cana-1458 140 42 𝐡′)𝑛 𝐡′)𝑛 PROPN cana-1458 140 43 } } PUNCT cana-1458 140 44 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 140 45 { { PUNCT cana-1458 140 46 ( ( PUNCT cana-1458 140 47 𝐺 𝐺 NOUN cana-1458 140 48 , , PUNCT cana-1458 140 49 𝐡)𝑛 𝐡)𝑛 NUM cana-1458 140 50 } } PUNCT cana-1458 140 51 . . PUNCT cana-1458 141 1 since since SCONJ cana-1458 141 2 { { PUNCT cana-1458 141 3 ( ( PUNCT cana-1458 141 4 𝐺 𝐺 NOUN cana-1458 141 5 , , PUNCT cana-1458 141 6 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 141 7 } } PUNCT cana-1458 141 8 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 141 9 { { PUNCT cana-1458 141 10 ( ( PUNCT cana-1458 141 11 𝐻 𝐻 PROPN cana-1458 141 12 , , PUNCT cana-1458 141 13 𝐢)𝑛 𝐢)𝑛 NUM cana-1458 141 14 } } PUNCT cana-1458 141 15 , , PUNCT cana-1458 141 16 { { PUNCT cana-1458 141 17 ( ( PUNCT cana-1458 141 18 𝐹 𝐹 PROPN cana-1458 141 19 , , PUNCT cana-1458 141 20 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 141 21 } } PUNCT cana-1458 141 22 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 141 23 { { PUNCT cana-1458 141 24 ( ( PUNCT cana-1458 141 25 𝐺′ 𝐺′ PROPN cana-1458 141 26 , , PUNCT cana-1458 141 27 𝐡′)𝑛 𝐡′)𝑛 PROPN cana-1458 141 28 } } PUNCT cana-1458 141 29 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 141 30 { { PUNCT cana-1458 141 31 ( ( PUNCT cana-1458 141 32 𝐻 𝐻 PROPN cana-1458 141 33 , , PUNCT cana-1458 141 34 𝐢)𝑛 𝐢)𝑛 NUM cana-1458 141 35 } } PUNCT cana-1458 141 36 . . PUNCT cana-1458 142 1 hence hence ADV cana-1458 142 2 { { PUNCT cana-1458 142 3 ( ( PUNCT cana-1458 142 4 𝐻 𝐻 PROPN cana-1458 142 5 , , PUNCT cana-1458 142 6 𝐢)𝑛 𝐢)𝑛 NUM cana-1458 142 7 } } PUNCT cana-1458 142 8 ∈ ∈ PROPN cana-1458 142 9 𝒩 𝒩 PROPN cana-1458 142 10 { { PUNCT cana-1458 142 11 ( ( PUNCT cana-1458 142 12 𝐹,𝐴)𝑛 𝐹,𝐴)𝑛 ADJ cana-1458 142 13 } } PUNCT cana-1458 142 14 . . PUNCT cana-1458 143 1 definition definition NOUN cana-1458 143 2 4.4 4.4 NUM cana-1458 143 3 . . PUNCT cana-1458 144 1 fuzzy fuzzy ADJ cana-1458 144 2 soft soft ADJ cana-1458 144 3 sequential sequential ADJ cana-1458 144 4 sets set NOUN cana-1458 144 5 { { PUNCT cana-1458 144 6 ( ( PUNCT cana-1458 144 7 𝐹 𝐹 PROPN cana-1458 144 8 , , PUNCT cana-1458 144 9 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 144 10 } } PUNCT cana-1458 144 11 and and CCONJ cana-1458 144 12 { { PUNCT cana-1458 144 13 ( ( PUNCT cana-1458 144 14 𝐺 𝐺 NOUN cana-1458 144 15 , , PUNCT cana-1458 144 16 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 144 17 } } PUNCT cana-1458 144 18 are be AUX cana-1458 144 19 said say VERB cana-1458 144 20 to to PART cana-1458 144 21 be be AUX cana-1458 144 22 fuzzy fuzzy ADJ cana-1458 144 23 soft soft ADJ cana-1458 144 24 sequential sequential ADJ cana-1458 144 25 quasi quasi NOUN cana-1458 144 26 - - NOUN cana-1458 144 27 coincident coincident ADJ cana-1458 144 28 , , PUNCT cana-1458 144 29 denoted denote VERB cana-1458 144 30 by by ADP cana-1458 144 31 { { PUNCT cana-1458 144 32 ( ( PUNCT cana-1458 144 33 𝐹 𝐹 PROPN cana-1458 144 34 , , PUNCT cana-1458 144 35 𝐴)𝑛}π‘ž{(𝐺 𝐴)𝑛}π‘ž{(𝐺 NOUN cana-1458 144 36 , , PUNCT cana-1458 144 37 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 144 38 } } PUNCT cana-1458 144 39 , , PUNCT cana-1458 144 40 if if SCONJ cana-1458 144 41 and and CCONJ cana-1458 144 42 only only ADV cana-1458 144 43 if if SCONJ cana-1458 144 44 there there PRON cana-1458 144 45 exists exist VERB cana-1458 144 46 𝑒𝑖 𝑒𝑖 ADP cana-1458 144 47 ∈ ∈ PROPN cana-1458 144 48 𝐴 𝐴 PROPN cana-1458 144 49 , , PUNCT cana-1458 144 50 𝐡 𝐡 NOUN cana-1458 144 51 and and CCONJ cana-1458 144 52 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 144 53 ∈ ∈ PROPN cana-1458 144 54 π‘ˆ π‘ˆ PROPN cana-1458 144 55 such such ADJ cana-1458 144 56 that that DET cana-1458 144 57 𝐹n(𝑒𝑖)(π‘₯𝑗 𝐹n(𝑒𝑖)(π‘₯𝑗 NUM cana-1458 144 58 ) ) PUNCT cana-1458 144 59 + + CCONJ cana-1458 144 60 𝐺n(𝑒𝑖)(π‘₯𝑗 𝐺n(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 144 61 ) ) PUNCT cana-1458 144 62 > > PUNCT cana-1458 144 63 Μƒ Μƒ NOUN cana-1458 144 64 1 1 NUM cana-1458 144 65 ( ( PUNCT cana-1458 144 66 π‘œπ‘Ÿ)𝐹n(𝑒𝑖)(π‘₯𝑗 π‘œπ‘Ÿ)𝐹n(𝑒𝑖)(π‘₯𝑗 PROPN cana-1458 144 67 ) ) PUNCT cana-1458 144 68 > > PUNCT cana-1458 144 69 Μƒ Μƒ ADJ cana-1458 144 70 𝐺n 𝐺n PROPN cana-1458 144 71 c(𝑒𝑖)(π‘₯𝑗 c(𝑒𝑖)(π‘₯𝑗 PROPN cana-1458 144 72 ) ) PUNCT cana-1458 144 73 for for ADP cana-1458 144 74 all all DET cana-1458 144 75 𝑛 𝑛 DET cana-1458 144 76 ∈ ∈ NOUN cana-1458 144 77 β„•. β„•. PROPN cana-1458 144 78 if if SCONJ cana-1458 144 79 there there PRON cana-1458 144 80 exists exist VERB cana-1458 144 81 𝑒𝑖 𝑒𝑖 ADP cana-1458 144 82 ∈ ∈ PROPN cana-1458 144 83 𝐴 𝐴 PROPN cana-1458 144 84 , , PUNCT cana-1458 144 85 𝐡 𝐡 NOUN cana-1458 144 86 and and CCONJ cana-1458 144 87 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 144 88 ∈ ∈ PROPN cana-1458 144 89 π‘ˆ π‘ˆ PROPN cana-1458 144 90 such such ADJ cana-1458 144 91 that that DET cana-1458 144 92 𝐹n(𝑒𝑖)(π‘₯𝑗 𝐹n(𝑒𝑖)(π‘₯𝑗 NUM cana-1458 144 93 ) ) PUNCT cana-1458 144 94 + + CCONJ cana-1458 144 95 𝐺n(𝑒𝑖)(π‘₯𝑗 𝐺n(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 144 96 ) ) PUNCT cana-1458 144 97 ≀̃ ≀̃ NOUN cana-1458 144 98 1 1 NUM cana-1458 144 99 for for ADP cana-1458 144 100 all all DET cana-1458 144 101 𝑛 𝑛 DET cana-1458 144 102 ∈ ∈ PROPN cana-1458 144 103 β„• β„• PROPN cana-1458 144 104 , , PUNCT cana-1458 144 105 then then ADV cana-1458 144 106 { { PUNCT cana-1458 144 107 ( ( PUNCT cana-1458 144 108 𝐹 𝐹 PROPN cana-1458 144 109 , , PUNCT cana-1458 144 110 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 144 111 } } PUNCT cana-1458 144 112 and and CCONJ cana-1458 144 113 { { PUNCT cana-1458 144 114 ( ( PUNCT cana-1458 144 115 𝐺 𝐺 NOUN cana-1458 144 116 , , PUNCT cana-1458 144 117 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 144 118 } } PUNCT cana-1458 144 119 are be AUX cana-1458 144 120 not not PART cana-1458 144 121 fuzzy fuzzy ADJ cana-1458 144 122 soft soft ADJ cana-1458 144 123 sequential sequential ADJ cana-1458 144 124 quasi quasi NOUN cana-1458 144 125 - - NOUN cana-1458 144 126 coincident coincident ADJ cana-1458 145 1 and and CCONJ cana-1458 145 2 it it PRON cana-1458 145 3 is be AUX cana-1458 145 4 denoted denote VERB cana-1458 145 5 by by ADP cana-1458 145 6 { { PUNCT cana-1458 145 7 ( ( PUNCT cana-1458 145 8 𝐹 𝐹 PROPN cana-1458 145 9 , , PUNCT cana-1458 145 10 𝐴)𝑛} 𝐴)𝑛} NOUN cana-1458 145 11 οΏ½ οΏ½ NOUN cana-1458 145 12 Μ… Μ… NOUN cana-1458 145 13 οΏ½ οΏ½ NOUN cana-1458 145 14 {(𝐺 {(𝐺 NOUN cana-1458 145 15 , , PUNCT cana-1458 145 16 𝐡)𝑛 𝐡)𝑛 NUM cana-1458 145 17 } } PUNCT cana-1458 145 18 . . PUNCT cana-1458 146 1 definition definition NOUN cana-1458 146 2 4.5 4.5 NUM cana-1458 146 3 . . PUNCT cana-1458 147 1 fuzzy fuzzy ADJ cana-1458 147 2 soft soft ADJ cana-1458 147 3 sequential sequential ADJ cana-1458 147 4 sets set NOUN cana-1458 147 5 { { PUNCT cana-1458 147 6 ( ( PUNCT cana-1458 147 7 𝐹 𝐹 PROPN cana-1458 147 8 , , PUNCT cana-1458 147 9 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 147 10 } } PUNCT cana-1458 147 11 and and CCONJ cana-1458 147 12 { { PUNCT cana-1458 147 13 ( ( PUNCT cana-1458 147 14 𝐺 𝐺 NOUN cana-1458 147 15 , , PUNCT cana-1458 147 16 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 147 17 } } PUNCT cana-1458 147 18 are be AUX cana-1458 147 19 said say VERB cana-1458 147 20 to to PART cana-1458 147 21 be be AUX cana-1458 147 22 fuzzy fuzzy ADJ cana-1458 147 23 soft soft ADJ cana-1458 147 24 sequential sequential ADJ cana-1458 147 25 weakly weakly ADJ cana-1458 147 26 quasi quasi NOUN cana-1458 147 27 - - NOUN cana-1458 147 28 coincident coincident ADJ cana-1458 147 29 , , PUNCT cana-1458 147 30 denoted denote VERB cana-1458 147 31 by by ADP cana-1458 147 32 { { PUNCT cana-1458 147 33 ( ( PUNCT cana-1458 147 34 𝐹 𝐹 PROPN cana-1458 147 35 , , PUNCT cana-1458 147 36 𝐴)𝑛}π‘žπ‘€{(𝐺 𝐴)𝑛}π‘žπ‘€{(𝐺 PROPN cana-1458 147 37 , , PUNCT cana-1458 147 38 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 147 39 } } PUNCT cana-1458 147 40 , , PUNCT cana-1458 147 41 if if SCONJ cana-1458 147 42 and and CCONJ cana-1458 147 43 only only ADV cana-1458 147 44 if if SCONJ cana-1458 147 45 there there PRON cana-1458 147 46 exists exist VERB cana-1458 147 47 𝑒𝑖 𝑒𝑖 ADP cana-1458 147 48 ∈ ∈ PROPN cana-1458 147 49 𝐴 𝐴 PROPN cana-1458 147 50 , , PUNCT cana-1458 147 51 𝐡 𝐡 NOUN cana-1458 147 52 and and CCONJ cana-1458 147 53 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 147 54 ∈ ∈ PROPN cana-1458 147 55 π‘ˆ π‘ˆ PROPN cana-1458 147 56 such such ADJ cana-1458 147 57 that that DET cana-1458 147 58 𝐹n(𝑒𝑖)(π‘₯𝑗 𝐹n(𝑒𝑖)(π‘₯𝑗 NUM cana-1458 147 59 ) ) PUNCT cana-1458 147 60 + + CCONJ cana-1458 147 61 𝐺n(𝑒𝑖)(π‘₯𝑗 𝐺n(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 147 62 ) ) PUNCT cana-1458 147 63 > > PUNCT cana-1458 147 64 Μƒ Μƒ NOUN cana-1458 147 65 1 1 NUM cana-1458 147 66 ( ( PUNCT cana-1458 147 67 π‘œπ‘Ÿ)𝐹n(𝑒𝑖)(π‘₯𝑗 π‘œπ‘Ÿ)𝐹n(𝑒𝑖)(π‘₯𝑗 PROPN cana-1458 147 68 ) ) PUNCT cana-1458 147 69 > > PUNCT cana-1458 147 70 Μƒ Μƒ ADJ cana-1458 147 71 𝐺n 𝐺n PROPN cana-1458 147 72 c(𝑒𝑖)(π‘₯𝑗 c(𝑒𝑖)(π‘₯𝑗 PROPN cana-1458 147 73 ) ) PUNCT cana-1458 147 74 for for ADP cana-1458 147 75 some some DET cana-1458 147 76 𝑛 𝑛 DET cana-1458 147 77 ∈ ∈ PROPN cana-1458 147 78 β„•. β„•. PROPN cana-1458 147 79 if if SCONJ cana-1458 147 80 { { PUNCT cana-1458 147 81 ( ( PUNCT cana-1458 147 82 𝐹 𝐹 PROPN cana-1458 147 83 , , PUNCT cana-1458 147 84 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 147 85 } } PUNCT cana-1458 147 86 and and CCONJ cana-1458 147 87 { { PUNCT cana-1458 147 88 ( ( PUNCT cana-1458 147 89 𝐺 𝐺 NOUN cana-1458 147 90 , , PUNCT cana-1458 147 91 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 147 92 } } PUNCT cana-1458 147 93 are be AUX cana-1458 147 94 not not PART cana-1458 147 95 fuzzy fuzzy ADJ cana-1458 147 96 soft soft ADJ cana-1458 147 97 sequential sequential ADJ cana-1458 147 98 weakly weakly ADJ cana-1458 147 99 quasi quasi NOUN cana-1458 147 100 - - NOUN cana-1458 147 101 coincident coincident ADJ cana-1458 147 102 , , PUNCT cana-1458 147 103 then then ADV cana-1458 147 104 it it PRON cana-1458 147 105 can can AUX cana-1458 147 106 be be AUX cana-1458 147 107 written write VERB cana-1458 147 108 as as ADP cana-1458 147 109 { { PUNCT cana-1458 147 110 ( ( PUNCT cana-1458 147 111 𝐹 𝐹 PROPN cana-1458 147 112 , , PUNCT cana-1458 147 113 𝐴)𝑛}π‘žπ‘€Μ…Μ…Μ…Μ… 𝐴)𝑛}π‘žπ‘€Μ…Μ…Μ…Μ… PROPN cana-1458 147 114 { { PUNCT cana-1458 147 115 ( ( PUNCT cana-1458 147 116 𝐺 𝐺 NOUN cana-1458 147 117 , , PUNCT cana-1458 147 118 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 147 119 } } PUNCT cana-1458 147 120 . . PUNCT cana-1458 148 1 proposition proposition NOUN cana-1458 148 2 4.6 4.6 NUM cana-1458 148 3 the the DET cana-1458 148 4 fuzzy fuzzy ADJ cana-1458 148 5 soft soft ADJ cana-1458 148 6 sequential sequential ADJ cana-1458 148 7 sets set NOUN cana-1458 148 8 { { PUNCT cana-1458 148 9 ( ( PUNCT cana-1458 148 10 𝐹 𝐹 PROPN cana-1458 148 11 , , PUNCT cana-1458 148 12 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 148 13 } } PUNCT cana-1458 148 14 and and CCONJ cana-1458 148 15 { { PUNCT cana-1458 148 16 ( ( PUNCT cana-1458 148 17 𝐺 𝐺 NOUN cana-1458 148 18 , , PUNCT cana-1458 148 19 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 148 20 } } PUNCT cana-1458 148 21 are be AUX cana-1458 148 22 fuzzy fuzzy ADJ cana-1458 148 23 soft soft ADJ cana-1458 148 24 sequential sequential ADJ cana-1458 148 25 quasi quasi NOUN cana-1458 148 26 - - NOUN cana-1458 148 27 coincident coincident ADJ cana-1458 148 28 , , PUNCT cana-1458 148 29 if if SCONJ cana-1458 148 30 and and CCONJ cana-1458 148 31 only only ADV cana-1458 148 32 if if SCONJ cana-1458 148 33 each each DET cana-1458 148 34 pair pair NOUN cana-1458 148 35 of of ADP cana-1458 148 36 non non ADJ cana-1458 148 37 - - ADJ cana-1458 148 38 zero zero ADJ cana-1458 148 39 fuzzy fuzzy ADJ cana-1458 148 40 soft soft ADJ cana-1458 148 41 sets set NOUN cana-1458 148 42 ( ( PUNCT cana-1458 148 43 𝐹 𝐹 PROPN cana-1458 148 44 , , PUNCT cana-1458 148 45 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 148 46 and and CCONJ cana-1458 148 47 ( ( PUNCT cana-1458 148 48 𝐺 𝐺 PROPN cana-1458 148 49 , , PUNCT cana-1458 148 50 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 148 51 are be AUX cana-1458 148 52 also also ADV cana-1458 148 53 so so ADV cana-1458 148 54 . . PUNCT cana-1458 149 1 proof proof NOUN cana-1458 149 2 . . PUNCT cana-1458 150 1 let let VERB cana-1458 150 2 the the DET cana-1458 150 3 fuzzy fuzzy ADJ cana-1458 150 4 soft soft ADJ cana-1458 150 5 sequential sequential ADJ cana-1458 150 6 sets set NOUN cana-1458 150 7 { { PUNCT cana-1458 150 8 ( ( PUNCT cana-1458 150 9 𝐹 𝐹 PROPN cana-1458 150 10 , , PUNCT cana-1458 150 11 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 150 12 } } PUNCT cana-1458 150 13 and and CCONJ cana-1458 150 14 { { PUNCT cana-1458 150 15 ( ( PUNCT cana-1458 150 16 𝐺 𝐺 NOUN cana-1458 150 17 , , PUNCT cana-1458 150 18 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 150 19 } } PUNCT cana-1458 150 20 are be AUX cana-1458 150 21 fuzzy fuzzy ADJ cana-1458 150 22 soft soft ADJ cana-1458 150 23 sequential sequential ADJ cana-1458 150 24 quasicoincident quasicoincident ADJ cana-1458 150 25 ⇔ ⇔ PROPN cana-1458 150 26 there there PRON cana-1458 150 27 exists exist VERB cana-1458 150 28 𝑒𝑖 𝑒𝑖 ADP cana-1458 150 29 ∈ ∈ PROPN cana-1458 150 30 𝐴 𝐴 PROPN cana-1458 150 31 , , PUNCT cana-1458 150 32 b b PROPN cana-1458 150 33 and and CCONJ cana-1458 150 34 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 150 35 ∈ ∈ PROPN cana-1458 150 36 π‘ˆ π‘ˆ PROPN cana-1458 150 37 such such ADJ cana-1458 150 38 that that SCONJ cana-1458 150 39 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 150 40 ) ) PUNCT cana-1458 151 1 + + PUNCT cana-1458 151 2 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 151 3 ) ) PUNCT cana-1458 152 1 > > PUNCT cana-1458 152 2 Μƒ Μƒ NOUN cana-1458 152 3 1 1 NUM cana-1458 152 4 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-1458 152 5 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ VERB cana-1458 152 6 𝑛 𝑛 DET cana-1458 152 7 ∈ ∈ PROPN cana-1458 152 8 β„• β„• PROPN cana-1458 152 9 communications communication NOUN cana-1458 152 10 on on ADP cana-1458 152 11 applied apply VERB cana-1458 152 12 nonlinear nonlinear ADJ cana-1458 152 13 analysis analysis NOUN cana-1458 152 14 issn issn NOUN cana-1458 152 15 : : PUNCT cana-1458 152 16 1074 1074 NUM cana-1458 152 17 - - PUNCT cana-1458 152 18 133x 133x NUM cana-1458 152 19 vol vol NOUN cana-1458 152 20 31 31 NUM cana-1458 152 21 no no NOUN cana-1458 152 22 . . PUNCT cana-1458 153 1 8s 8s PROPN cana-1458 153 2 ( ( PUNCT cana-1458 153 3 2024 2024 NUM cana-1458 153 4 ) ) PUNCT cana-1458 153 5 93 93 NUM cana-1458 153 6 https://internationalpubls.com https://internationalpubls.com X cana-1458 153 7 ⇔ ⇔ NOUN cana-1458 153 8 each each DET cana-1458 153 9 pair pair NOUN cana-1458 153 10 of of ADP cana-1458 153 11 non non ADJ cana-1458 153 12 - - ADJ cana-1458 153 13 zero zero ADJ cana-1458 153 14 fuzzy fuzzy ADJ cana-1458 153 15 soft soft ADJ cana-1458 153 16 sets set NOUN cana-1458 153 17 ( ( PUNCT cana-1458 153 18 𝐹 𝐹 PROPN cana-1458 153 19 , , PUNCT cana-1458 153 20 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 153 21 and and CCONJ cana-1458 153 22 ( ( PUNCT cana-1458 153 23 𝐺 𝐺 PROPN cana-1458 153 24 , , PUNCT cana-1458 153 25 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 153 26 are be AUX cana-1458 153 27 also also ADV cana-1458 153 28 fuzzy fuzzy ADJ cana-1458 153 29 soft soft ADJ cana-1458 153 30 quasi quasi NOUN cana-1458 153 31 - - ADJ cana-1458 153 32 coincident coincident ADJ cana-1458 153 33 . . PUNCT cana-1458 154 1 proposition proposition NOUN cana-1458 154 2 4.7 4.7 NUM cana-1458 154 3 . . PUNCT cana-1458 155 1 let let VERB cana-1458 155 2 { { PUNCT cana-1458 155 3 ( ( PUNCT cana-1458 155 4 𝐹 𝐹 PROPN cana-1458 155 5 , , PUNCT cana-1458 155 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 155 7 } } PUNCT cana-1458 155 8 and and CCONJ cana-1458 155 9 { { PUNCT cana-1458 155 10 ( ( PUNCT cana-1458 155 11 𝐺 𝐺 NOUN cana-1458 155 12 , , PUNCT cana-1458 155 13 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 155 14 } } PUNCT cana-1458 155 15 be be AUX cana-1458 155 16 a a DET cana-1458 155 17 fuzzy fuzzy ADJ cana-1458 155 18 soft soft ADJ cana-1458 155 19 sequential sequential ADJ cana-1458 155 20 sets set NOUN cana-1458 155 21 over over ADP cana-1458 155 22 π‘ˆ. π‘ˆ. PROPN cana-1458 155 23 then then ADV cana-1458 155 24 the the DET cana-1458 155 25 following follow VERB cana-1458 155 26 properties property NOUN cana-1458 155 27 are be AUX cana-1458 155 28 true true ADJ cana-1458 155 29 . . PUNCT cana-1458 156 1 ( ( PUNCT cana-1458 156 2 i i NOUN cana-1458 156 3 ) ) PUNCT cana-1458 156 4 { { PUNCT cana-1458 156 5 ( ( PUNCT cana-1458 156 6 𝐹 𝐹 PROPN cana-1458 156 7 , , PUNCT cana-1458 156 8 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 156 9 } } PUNCT cana-1458 156 10 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 156 11 { { PUNCT cana-1458 156 12 ( ( PUNCT cana-1458 156 13 𝐺 𝐺 NOUN cana-1458 156 14 , , PUNCT cana-1458 156 15 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 156 16 } } PUNCT cana-1458 156 17 if if SCONJ cana-1458 156 18 and and CCONJ cana-1458 156 19 only only ADV cana-1458 156 20 if if SCONJ cana-1458 156 21 { { PUNCT cana-1458 156 22 ( ( PUNCT cana-1458 156 23 𝐹 𝐹 PROPN cana-1458 156 24 , , PUNCT cana-1458 156 25 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 156 26 } } PUNCT cana-1458 156 27 οΏ½ οΏ½ PROPN cana-1458 156 28 Μ… Μ… NOUN cana-1458 156 29 οΏ½ οΏ½ NOUN cana-1458 156 30 {(𝐺 {(𝐺 SYM cana-1458 156 31 , , PUNCT cana-1458 156 32 𝐡)𝑛}𝑐 𝐡)𝑛}𝑐 NOUN cana-1458 156 33 ( ( PUNCT cana-1458 156 34 ii ii NOUN cana-1458 156 35 ) ) PUNCT cana-1458 156 36 { { PUNCT cana-1458 156 37 ( ( PUNCT cana-1458 156 38 𝐹 𝐹 PROPN cana-1458 156 39 , , PUNCT cana-1458 156 40 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 156 41 } } PUNCT cana-1458 156 42 βŠ‚Μƒπ‘€ βŠ‚Μƒπ‘€ NOUN cana-1458 156 43 { { PUNCT cana-1458 156 44 ( ( PUNCT cana-1458 156 45 𝐺 𝐺 NOUN cana-1458 156 46 , , PUNCT cana-1458 156 47 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 156 48 } } PUNCT cana-1458 156 49 if if SCONJ cana-1458 156 50 and and CCONJ cana-1458 156 51 only only ADV cana-1458 156 52 if if SCONJ cana-1458 156 53 { { PUNCT cana-1458 156 54 ( ( PUNCT cana-1458 156 55 𝐹 𝐹 PROPN cana-1458 156 56 , , PUNCT cana-1458 156 57 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 156 58 } } PUNCT cana-1458 156 59 π‘žπ‘€{(𝐺 π‘žπ‘€{(𝐺 NOUN cana-1458 156 60 , , PUNCT cana-1458 156 61 𝐡)𝑛}𝑐 𝐡)𝑛}𝑐 NOUN cana-1458 156 62 ( ( PUNCT cana-1458 156 63 iii iii NOUN cana-1458 156 64 ) ) PUNCT cana-1458 156 65 { { PUNCT cana-1458 156 66 ( ( PUNCT cana-1458 156 67 𝐹 𝐹 PROPN cana-1458 156 68 , , PUNCT cana-1458 156 69 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 156 70 } } PUNCT cana-1458 156 71 βŠ„Μƒ βŠ„Μƒ X cana-1458 156 72 { { PUNCT cana-1458 156 73 ( ( PUNCT cana-1458 156 74 𝐺 𝐺 NOUN cana-1458 156 75 , , PUNCT cana-1458 156 76 𝐡)𝑛}𝑐 𝐡)𝑛}𝑐 NOUN cana-1458 156 77 if if SCONJ cana-1458 156 78 and and CCONJ cana-1458 156 79 only only ADV cana-1458 156 80 if if SCONJ cana-1458 156 81 { { PUNCT cana-1458 156 82 ( ( PUNCT cana-1458 156 83 𝐹 𝐹 PROPN cana-1458 156 84 , , PUNCT cana-1458 156 85 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 156 86 } } PUNCT cana-1458 156 87 π‘ž{(𝐺 π‘ž{(𝐺 PROPN cana-1458 156 88 , , PUNCT cana-1458 156 89 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 156 90 } } PUNCT cana-1458 156 91 ( ( PUNCT cana-1458 156 92 iv iv X cana-1458 156 93 ) ) PUNCT cana-1458 156 94 { { PUNCT cana-1458 156 95 ( ( PUNCT cana-1458 156 96 𝐹 𝐹 PROPN cana-1458 156 97 , , PUNCT cana-1458 156 98 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 156 99 } } PUNCT cana-1458 156 100 π‘ž π‘ž X cana-1458 156 101 { { PUNCT cana-1458 156 102 ( ( PUNCT cana-1458 156 103 𝐺 𝐺 NOUN cana-1458 156 104 , , PUNCT cana-1458 156 105 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 156 106 } } PUNCT cana-1458 156 107 implies imply VERB cana-1458 156 108 { { PUNCT cana-1458 156 109 ( ( PUNCT cana-1458 156 110 𝐹 𝐹 PROPN cana-1458 156 111 , , PUNCT cana-1458 156 112 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 156 113 } } PUNCT cana-1458 156 114 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 156 115 { { PUNCT cana-1458 156 116 ( ( PUNCT cana-1458 156 117 𝐺 𝐺 NOUN cana-1458 156 118 , , PUNCT cana-1458 156 119 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 156 120 } } PUNCT cana-1458 156 121 β‰  β‰  PROPN cana-1458 156 122 βˆ…Μƒ βˆ…Μƒ NOUN cana-1458 156 123 ( ( PUNCT cana-1458 156 124 v v NOUN cana-1458 156 125 ) ) PUNCT cana-1458 156 126 { { PUNCT cana-1458 156 127 ( ( PUNCT cana-1458 156 128 𝐹 𝐹 PROPN cana-1458 156 129 , , PUNCT cana-1458 156 130 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 156 131 } } PUNCT cana-1458 156 132 οΏ½ οΏ½ PROPN cana-1458 156 133 Μ… Μ… NOUN cana-1458 156 134 οΏ½ οΏ½ PROPN cana-1458 156 135 { { PUNCT cana-1458 156 136 ( ( PUNCT cana-1458 156 137 𝐹 𝐹 PROPN cana-1458 156 138 , , PUNCT cana-1458 156 139 𝐴)𝑛}𝑐 𝐴)𝑛}𝑐 NOUN cana-1458 156 140 proof proof NOUN cana-1458 156 141 . . PUNCT cana-1458 157 1 ( ( PUNCT cana-1458 157 2 i i NOUN cana-1458 157 3 ) ) PUNCT cana-1458 157 4 { { PUNCT cana-1458 157 5 ( ( PUNCT cana-1458 157 6 𝐹 𝐹 PROPN cana-1458 157 7 , , PUNCT cana-1458 157 8 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 157 9 } } PUNCT cana-1458 157 10 βŠ‚Μƒ βŠ‚Μƒ NOUN cana-1458 157 11 { { PUNCT cana-1458 157 12 ( ( PUNCT cana-1458 157 13 𝐺 𝐺 NOUN cana-1458 157 14 , , PUNCT cana-1458 157 15 𝐡)𝑛 𝐡)𝑛 NUM cana-1458 157 16 } } PUNCT cana-1458 157 17 ⇔ ⇔ NOUN cana-1458 157 18 for for ADP cana-1458 157 19 each each DET cana-1458 157 20 𝑒𝑖 𝑒𝑖 PRON cana-1458 157 21 ∈ ∈ PROPN cana-1458 157 22 𝐴 𝐴 PROPN cana-1458 157 23 , , PUNCT cana-1458 157 24 𝐡 𝐡 NOUN cana-1458 157 25 and and CCONJ cana-1458 157 26 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 157 27 ∈ ∈ PROPN cana-1458 157 28 π‘ˆ π‘ˆ PROPN cana-1458 157 29 such such ADJ cana-1458 157 30 that that SCONJ cana-1458 157 31 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 157 32 ) ) PUNCT cana-1458 157 33 ≀̃ ≀̃ NOUN cana-1458 157 34 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 157 35 ) ) PUNCT cana-1458 157 36 for for ADP cana-1458 157 37 all all DET cana-1458 157 38 𝑛 𝑛 DET cana-1458 157 39 ∈ ∈ PROPN cana-1458 157 40 β„•. β„•. PROPN cana-1458 157 41 ⇔ ⇔ PROPN cana-1458 157 42 for for ADP cana-1458 157 43 each each DET cana-1458 157 44 𝑒𝑖 𝑒𝑖 PRON cana-1458 157 45 ∈ ∈ PROPN cana-1458 157 46 𝐴 𝐴 PROPN cana-1458 157 47 , , PUNCT cana-1458 157 48 𝐡 𝐡 NOUN cana-1458 157 49 and and CCONJ cana-1458 157 50 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 157 51 ∈ ∈ PROPN cana-1458 157 52 π‘ˆ π‘ˆ PROPN cana-1458 157 53 such such ADJ cana-1458 157 54 that that SCONJ cana-1458 157 55 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 157 56 ) ) PUNCT cana-1458 157 57 + + CCONJ cana-1458 157 58 1 1 NUM cana-1458 157 59 βˆ’ βˆ’ NUM cana-1458 157 60 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 157 61 ) ) PUNCT cana-1458 157 62 ≀̃ ≀̃ NOUN cana-1458 157 63 1 1 NUM cana-1458 157 64 for for ADP cana-1458 157 65 all all DET cana-1458 157 66 𝑛 𝑛 DET cana-1458 157 67 ∈ ∈ PROPN cana-1458 157 68 β„•. β„•. PROPN cana-1458 157 69 ⇔ ⇔ PROPN cana-1458 157 70 for for ADP cana-1458 157 71 each each DET cana-1458 157 72 𝑒𝑖 𝑒𝑖 PRON cana-1458 157 73 ∈ ∈ PROPN cana-1458 157 74 𝐴 𝐴 PROPN cana-1458 157 75 , , PUNCT cana-1458 157 76 𝐡 𝐡 NOUN cana-1458 157 77 and and CCONJ cana-1458 157 78 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 157 79 ∈ ∈ PROPN cana-1458 157 80 π‘ˆ π‘ˆ PROPN cana-1458 157 81 such such ADJ cana-1458 157 82 that that SCONJ cana-1458 157 83 { { PUNCT cana-1458 157 84 ( ( PUNCT cana-1458 157 85 𝐹 𝐹 PROPN cana-1458 157 86 , , PUNCT cana-1458 157 87 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 157 88 } } PUNCT cana-1458 157 89 + + CCONJ cana-1458 157 90 { { PUNCT cana-1458 157 91 ( ( PUNCT cana-1458 157 92 𝐺 𝐺 NOUN cana-1458 157 93 , , PUNCT cana-1458 157 94 𝐡)𝑛}𝑐 𝐡)𝑛}𝑐 NOUN cana-1458 157 95 ≀̃ ≀̃ NOUN cana-1458 157 96 1 1 NUM cana-1458 157 97 for for ADP cana-1458 157 98 all all DET cana-1458 157 99 𝑛 𝑛 DET cana-1458 157 100 ∈ ∈ NOUN cana-1458 157 101 β„•. β„•. PROPN cana-1458 157 102 hence hence ADV cana-1458 157 103 { { PUNCT cana-1458 157 104 ( ( PUNCT cana-1458 157 105 𝐹 𝐹 PROPN cana-1458 157 106 , , PUNCT cana-1458 157 107 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 157 108 } } PUNCT cana-1458 157 109 οΏ½ οΏ½ PROPN cana-1458 157 110 Μ… Μ… NOUN cana-1458 157 111 οΏ½ οΏ½ NOUN cana-1458 157 112 {(𝐺 {(𝐺 PUNCT cana-1458 157 113 , , PUNCT cana-1458 157 114 𝐡)𝑛}𝑐. 𝐡)𝑛}𝑐. PROPN cana-1458 157 115 ( ( PUNCT cana-1458 157 116 ii ii PROPN cana-1458 157 117 ) ) PUNCT cana-1458 157 118 the the DET cana-1458 157 119 proof proof NOUN cana-1458 157 120 is be AUX cana-1458 157 121 similar similar ADJ cana-1458 157 122 to to ADP cana-1458 157 123 ( ( PUNCT cana-1458 157 124 i i NOUN cana-1458 157 125 ) ) PUNCT cana-1458 157 126 ( ( PUNCT cana-1458 157 127 iii iii X cana-1458 157 128 ) ) PUNCT cana-1458 157 129 the the DET cana-1458 157 130 proof proof NOUN cana-1458 157 131 is be AUX cana-1458 157 132 similar similar ADJ cana-1458 157 133 to to ADP cana-1458 157 134 ( ( PUNCT cana-1458 157 135 i i NOUN cana-1458 157 136 ) ) PUNCT cana-1458 157 137 ( ( PUNCT cana-1458 157 138 iv iv X cana-1458 157 139 ) ) PUNCT cana-1458 157 140 let let VERB cana-1458 157 141 { { PUNCT cana-1458 157 142 ( ( PUNCT cana-1458 157 143 𝐹 𝐹 PROPN cana-1458 157 144 , , PUNCT cana-1458 157 145 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 157 146 } } PUNCT cana-1458 157 147 π‘ž π‘ž X cana-1458 157 148 { { PUNCT cana-1458 157 149 ( ( PUNCT cana-1458 157 150 𝐺 𝐺 NOUN cana-1458 157 151 , , PUNCT cana-1458 157 152 𝐡)𝑛 𝐡)𝑛 NUM cana-1458 157 153 } } PUNCT cana-1458 157 154 . . PUNCT cana-1458 158 1 then then ADV cana-1458 158 2 there there PRON cana-1458 158 3 exists exist VERB cana-1458 158 4 𝑒𝑖 𝑒𝑖 ADP cana-1458 158 5 ∈ ∈ PROPN cana-1458 158 6 𝐴 𝐴 PROPN cana-1458 158 7 , , PUNCT cana-1458 158 8 𝐡 𝐡 NOUN cana-1458 158 9 and and CCONJ cana-1458 158 10 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 158 11 ∈ ∈ PROPN cana-1458 158 12 π‘ˆ π‘ˆ PROPN cana-1458 158 13 such such ADJ cana-1458 158 14 that that SCONJ cana-1458 158 15 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 158 16 ) ) PUNCT cana-1458 158 17 + + PUNCT cana-1458 158 18 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 𝐺𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 158 19 ) ) PUNCT cana-1458 158 20 > > PUNCT cana-1458 158 21 Μƒ Μƒ NOUN cana-1458 158 22 1 1 NUM cana-1458 158 23 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-1458 158 24 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ VERB cana-1458 158 25 𝑛 𝑛 DET cana-1458 158 26 ∈ ∈ PROPN cana-1458 158 27 β„• β„• PROPN cana-1458 158 28 implies imply VERB cana-1458 158 29 { { PUNCT cana-1458 158 30 ( ( PUNCT cana-1458 158 31 𝐹 𝐹 PROPN cana-1458 158 32 , , PUNCT cana-1458 158 33 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 158 34 } } PUNCT cana-1458 158 35 β‰  β‰  PROPN cana-1458 158 36 βˆ…Μƒ βˆ…Μƒ NOUN cana-1458 158 37 and and CCONJ cana-1458 158 38 { { PUNCT cana-1458 158 39 ( ( PUNCT cana-1458 158 40 𝐺 𝐺 NOUN cana-1458 158 41 , , PUNCT cana-1458 158 42 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 158 43 } } PUNCT cana-1458 158 44 β‰  β‰  PROPN cana-1458 158 45 βˆ…Μƒ βˆ…Μƒ NOUN cana-1458 158 46 for for ADP cana-1458 158 47 all all DET cana-1458 158 48 𝑛 𝑛 DET cana-1458 158 49 ∈ ∈ NOUN cana-1458 158 50 β„•. β„•. PROPN cana-1458 158 51 hence hence ADV cana-1458 158 52 { { PUNCT cana-1458 158 53 ( ( PUNCT cana-1458 158 54 𝐹 𝐹 PROPN cana-1458 158 55 , , PUNCT cana-1458 158 56 𝐴)𝑛 𝐴)𝑛 VERB cana-1458 158 57 } } PUNCT cana-1458 158 58 βˆ©Μƒ βˆ©Μƒ PUNCT cana-1458 158 59 { { PUNCT cana-1458 158 60 ( ( PUNCT cana-1458 158 61 𝐺 𝐺 NOUN cana-1458 158 62 , , PUNCT cana-1458 158 63 𝐡)𝑛 𝐡)𝑛 NOUN cana-1458 158 64 } } PUNCT cana-1458 158 65 β‰  β‰  PROPN cana-1458 158 66 βˆ…Μƒ. βˆ…Μƒ. NOUN cana-1458 158 67 ( ( PUNCT cana-1458 158 68 v v NOUN cana-1458 158 69 ) ) PUNCT cana-1458 158 70 suppose suppose VERB cana-1458 158 71 { { PUNCT cana-1458 158 72 ( ( PUNCT cana-1458 158 73 𝐹 𝐹 PROPN cana-1458 158 74 , , PUNCT cana-1458 158 75 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 158 76 } } PUNCT cana-1458 158 77 π‘ž π‘ž X cana-1458 158 78 { { PUNCT cana-1458 158 79 ( ( PUNCT cana-1458 158 80 𝐹 𝐹 PROPN cana-1458 158 81 , , PUNCT cana-1458 158 82 𝐴)𝑛}𝑐. 𝐴)𝑛}𝑐. VERB cana-1458 158 83 then then ADV cana-1458 158 84 there there PRON cana-1458 158 85 exists exist VERB cana-1458 158 86 𝑒𝑖 𝑒𝑖 ADP cana-1458 158 87 ∈ ∈ PROPN cana-1458 158 88 𝐴 𝐴 PROPN cana-1458 158 89 and and CCONJ cana-1458 158 90 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 158 91 ∈ ∈ PROPN cana-1458 158 92 π‘ˆ π‘ˆ PROPN cana-1458 158 93 such such ADJ cana-1458 158 94 that that SCONJ cana-1458 158 95 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 158 96 ) ) PUNCT cana-1458 159 1 + + NUM cana-1458 159 2 𝐹𝑛 𝐹𝑛 PROPN cana-1458 159 3 c(𝑒𝑖)(π‘₯𝑗 c(𝑒𝑖)(π‘₯𝑗 PROPN cana-1458 159 4 ) ) PUNCT cana-1458 159 5 > > PUNCT cana-1458 160 1 Μƒ Μƒ NOUN cana-1458 160 2 1 1 NUM cana-1458 160 3 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-1458 160 4 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ VERB cana-1458 160 5 𝑛 𝑛 DET cana-1458 160 6 ∈ ∈ PROPN cana-1458 160 7 β„• β„• PROPN cana-1458 160 8 implies imply VERB cana-1458 160 9 there there PRON cana-1458 160 10 exists exist VERB cana-1458 160 11 𝑒𝑖 𝑒𝑖 ADP cana-1458 160 12 ∈ ∈ PROPN cana-1458 160 13 𝐴 𝐴 PROPN cana-1458 160 14 and and CCONJ cana-1458 160 15 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 160 16 ∈ ∈ PROPN cana-1458 160 17 π‘ˆ π‘ˆ PROPN cana-1458 160 18 such such ADJ cana-1458 160 19 that that SCONJ cana-1458 160 20 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 160 21 ) ) PUNCT cana-1458 161 1 + + CCONJ cana-1458 161 2 1 1 NUM cana-1458 161 3 βˆ’ βˆ’ NUM cana-1458 161 4 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 161 5 ) ) PUNCT cana-1458 161 6 > > PUNCT cana-1458 162 1 Μƒ Μƒ NOUN cana-1458 162 2 1 1 NUM cana-1458 162 3 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-1458 162 4 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ VERB cana-1458 162 5 𝑛 𝑛 DET cana-1458 162 6 ∈ ∈ PROPN cana-1458 162 7 β„•. β„•. PROPN cana-1458 162 8 therefore therefore ADV cana-1458 162 9 , , PUNCT cana-1458 162 10 there there PRON cana-1458 162 11 exists exist VERB cana-1458 162 12 𝑒𝑖 𝑒𝑖 ADP cana-1458 162 13 ∈ ∈ PROPN cana-1458 162 14 𝐴 𝐴 PROPN cana-1458 162 15 and and CCONJ cana-1458 162 16 π‘₯𝑗 π‘₯𝑗 PROPN cana-1458 162 17 ∈ ∈ PROPN cana-1458 162 18 π‘ˆ π‘ˆ PROPN cana-1458 162 19 such such ADJ cana-1458 162 20 that that SCONJ cana-1458 162 21 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 162 22 ) ) PUNCT cana-1458 162 23 > > PUNCT cana-1458 162 24 Μƒ Μƒ NOUN cana-1458 162 25 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 𝐹𝑛(𝑒𝑖)(π‘₯𝑗 NOUN cana-1458 162 26 ) ) PUNCT cana-1458 162 27 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-1458 162 28 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ VERB cana-1458 162 29 𝑛 𝑛 PRON cana-1458 162 30 ∈ ∈ PROPN cana-1458 162 31 β„• β„• PROPN cana-1458 162 32 , , PUNCT cana-1458 162 33 which which PRON cana-1458 162 34 is be AUX cana-1458 162 35 a a DET cana-1458 162 36 contradiction contradiction NOUN cana-1458 162 37 . . PUNCT cana-1458 163 1 hence hence ADV cana-1458 163 2 { { PUNCT cana-1458 163 3 ( ( PUNCT cana-1458 163 4 𝐹 𝐹 PROPN cana-1458 163 5 , , PUNCT cana-1458 163 6 𝐴)𝑛 𝐴)𝑛 NOUN cana-1458 163 7 } } PUNCT cana-1458 163 8 οΏ½ οΏ½ PROPN cana-1458 163 9 Μ… Μ… NOUN cana-1458 163 10 οΏ½ οΏ½ PROPN cana-1458 163 11 { { PUNCT cana-1458 163 12 ( ( PUNCT cana-1458 163 13 𝐹 𝐹 PROPN cana-1458 163 14 , , PUNCT cana-1458 163 15 𝐴)𝑛}𝑐. 𝐴)𝑛}𝑐. NOUN cana-1458 163 16 5 5 NUM cana-1458 163 17 . . PUNCT cana-1458 163 18 conclusion conclusion NOUN cana-1458 163 19 in in ADP cana-1458 163 20 this this DET cana-1458 163 21 paper paper NOUN cana-1458 163 22 , , PUNCT cana-1458 163 23 we we PRON cana-1458 163 24 have have AUX cana-1458 163 25 introduced introduce VERB cana-1458 163 26 fuzzy fuzzy ADJ cana-1458 163 27 soft soft ADJ cana-1458 163 28 sequential sequential ADJ cana-1458 163 29 topological topological ADJ cana-1458 163 30 spaces space NOUN cana-1458 163 31 via via ADP cana-1458 163 32 fuzzy fuzzy ADJ cana-1458 163 33 soft soft ADJ cana-1458 163 34 sequential sequential ADJ cana-1458 163 35 set set NOUN cana-1458 163 36 . . PUNCT cana-1458 164 1 we we PRON cana-1458 164 2 studied study VERB cana-1458 164 3 their their PRON cana-1458 164 4 properties property NOUN cana-1458 164 5 and and CCONJ cana-1458 164 6 a a DET cana-1458 164 7 relationship relationship NOUN cana-1458 164 8 between between ADP cana-1458 164 9 fuzzy fuzzy ADJ cana-1458 164 10 soft soft ADJ cana-1458 164 11 topology topology NOUN cana-1458 164 12 and and CCONJ cana-1458 164 13 fuzzy fuzzy ADJ cana-1458 164 14 soft soft ADJ cana-1458 164 15 sequential sequential ADJ cana-1458 164 16 topology topology NOUN cana-1458 164 17 are be AUX cana-1458 164 18 explored explore VERB cana-1458 164 19 . . PUNCT cana-1458 165 1 we we PRON cana-1458 165 2 investigated investigate VERB cana-1458 165 3 fuzzy fuzzy ADJ cana-1458 165 4 soft soft ADJ cana-1458 165 5 sequential sequential ADJ cana-1458 165 6 neighborhood neighborhood NOUN cana-1458 165 7 of of ADP cana-1458 165 8 a a DET cana-1458 165 9 fuzzy fuzzy ADJ cana-1458 165 10 soft soft ADJ cana-1458 165 11 sequential sequential ADJ cana-1458 165 12 set set NOUN cana-1458 165 13 , , PUNCT cana-1458 165 14 fuzzy fuzzy ADJ cana-1458 165 15 soft soft ADJ cana-1458 165 16 sequential sequential ADJ cana-1458 165 17 quasi quasi NOUN cana-1458 165 18 - - NOUN cana-1458 165 19 coincident coincident ADJ cana-1458 165 20 along along ADP cana-1458 165 21 with with ADP cana-1458 165 22 their their PRON cana-1458 165 23 properties property NOUN cana-1458 165 24 . . PUNCT cana-1458 166 1 references reference NOUN cana-1458 166 2 [ [ X cana-1458 166 3 1 1 X cana-1458 166 4 ] ] X cana-1458 166 5 ahmad ahmad PROPN cana-1458 166 6 b b PROPN cana-1458 166 7 , , PUNCT cana-1458 166 8 athar athar VERB cana-1458 166 9 kharal kharal ADJ cana-1458 166 10 2009,’on 2009,’on NUM cana-1458 166 11 fuzzy fuzzy ADJ cana-1458 166 12 soft soft ADJ cana-1458 166 13 sets set NOUN cana-1458 166 14 ’ ' PUNCT cana-1458 166 15 , , PUNCT cana-1458 166 16 advances advance NOUN cana-1458 166 17 in in ADP cana-1458 166 18 fuzzy fuzzy ADJ cana-1458 166 19 systems system NOUN cana-1458 166 20 , , PUNCT cana-1458 166 21 pp pp ADP cana-1458 166 22 1 1 NUM cana-1458 166 23 - - SYM cana-1458 166 24 6 6 NUM cana-1458 166 25 . . PUNCT cana-1458 167 1 [ [ X cana-1458 167 2 2 2 NUM cana-1458 167 3 ] ] X cana-1458 167 4 bose bose NOUN cana-1458 167 5 m m VERB cana-1458 167 6 k k NOUN cana-1458 167 7 , , PUNCT cana-1458 167 8 indrajit indrajit VERB cana-1458 167 9 lahiri lahiri PROPN cana-1458 167 10 2002 2002 NUM cana-1458 167 11 , , PUNCT cana-1458 167 12 β€˜ ' PUNCT cana-1458 167 13 sequential sequential ADJ cana-1458 167 14 topological topological ADJ cana-1458 167 15 spaces space NOUN cana-1458 167 16 and and CCONJ cana-1458 167 17 separation separation NOUN cana-1458 167 18 axioms axiom NOUN cana-1458 167 19 ’ ' PUNCT cana-1458 167 20 , , PUNCT cana-1458 167 21 bulletin bulletin NOUN cana-1458 167 22 of of ADP cana-1458 167 23 the the DET cana-1458 167 24 allahabad allahabad PROPN cana-1458 167 25 mathematical mathematical PROPN cana-1458 167 26 society society NOUN cana-1458 167 27 , , PUNCT cana-1458 167 28 17 17 NUM cana-1458 167 29 , , PUNCT cana-1458 167 30 pp pp ADV cana-1458 167 31 23 23 NUM cana-1458 167 32 - - SYM cana-1458 167 33 37 37 NUM cana-1458 167 34 . . PUNCT cana-1458 168 1 communications communication NOUN cana-1458 168 2 on on ADP cana-1458 168 3 applied apply VERB cana-1458 168 4 nonlinear nonlinear ADJ cana-1458 168 5 analysis analysis NOUN cana-1458 168 6 issn issn NOUN cana-1458 168 7 : : PUNCT cana-1458 168 8 1074 1074 NUM cana-1458 168 9 - - PUNCT cana-1458 168 10 133x 133x NUM cana-1458 168 11 vol vol NOUN cana-1458 168 12 31 31 NUM cana-1458 168 13 no no NOUN cana-1458 168 14 . . PUNCT cana-1458 169 1 8s 8s PROPN cana-1458 169 2 ( ( PUNCT cana-1458 169 3 2024 2024 NUM cana-1458 169 4 ) ) PUNCT cana-1458 169 5 94 94 NUM cana-1458 169 6 https://internationalpubls.com https://internationalpubls.com X cana-1458 170 1 [ [ X cana-1458 170 2 3 3 X cana-1458 170 3 ] ] X cana-1458 170 4 cagman cagman NOUN cana-1458 170 5 n n NOUN cana-1458 170 6 , , PUNCT cana-1458 170 7 enginoglu enginoglu PROPN cana-1458 170 8 s s PART cana-1458 170 9 and and CCONJ cana-1458 170 10 citak citak NOUN cana-1458 170 11 f f PROPN cana-1458 170 12 2011 2011 NUM cana-1458 170 13 , , PUNCT cana-1458 170 14 β€˜ ' PUNCT cana-1458 170 15 soft soft ADJ cana-1458 170 16 topology topology NOUN cana-1458 170 17 ’ ' PUNCT cana-1458 170 18 , , PUNCT cana-1458 170 19 computers computer NOUN cana-1458 170 20 and and CCONJ cana-1458 170 21 mathematics mathematic NOUN cana-1458 170 22 with with ADP cana-1458 170 23 applications,62 applications,62 NOUN cana-1458 170 24 , , PUNCT cana-1458 170 25 pp pp ADV cana-1458 170 26 351 351 NUM cana-1458 170 27 - - SYM cana-1458 170 28 358 358 NUM cana-1458 170 29 . . PUNCT cana-1458 171 1 [ [ X cana-1458 171 2 4 4 X cana-1458 171 3 ] ] X cana-1458 171 4 chang chang PROPN cana-1458 171 5 c c PROPN cana-1458 171 6 l l PROPN cana-1458 171 7 1968 1968 NUM cana-1458 171 8 , , PUNCT cana-1458 171 9 β€˜ ' PUNCT cana-1458 171 10 fuzzy fuzzy ADJ cana-1458 171 11 topological topological ADJ cana-1458 171 12 spaces space NOUN cana-1458 171 13 ’ ' PUNCT cana-1458 171 14 , , PUNCT cana-1458 171 15 journal journal NOUN cana-1458 171 16 of of ADP cana-1458 171 17 mathematical mathematical ADJ cana-1458 171 18 analysis analysis NOUN cana-1458 171 19 and and CCONJ cana-1458 171 20 applications application NOUN cana-1458 171 21 , , PUNCT cana-1458 171 22 vol.24 vol.24 NOUN cana-1458 171 23 , , PUNCT cana-1458 171 24 pp.182 pp.182 PROPN cana-1458 171 25 - - PUNCT cana-1458 171 26 290 290 NUM cana-1458 171 27 . . PUNCT cana-1458 172 1 [ [ X cana-1458 172 2 5 5 NUM cana-1458 172 3 ] ] X cana-1458 172 4 conrad conrad PROPN cana-1458 172 5 f f PROPN cana-1458 172 6 1980 1980 NUM cana-1458 172 7 , , PUNCT cana-1458 172 8 β€˜ ' PUNCT cana-1458 172 9 fuzzy fuzzy ADJ cana-1458 172 10 topological topological ADJ cana-1458 172 11 concepts concept NOUN cana-1458 172 12 ’ ' PUNCT cana-1458 172 13 , , PUNCT cana-1458 172 14 journal journal NOUN cana-1458 172 15 of of ADP cana-1458 172 16 mathematical mathematical ADJ cana-1458 172 17 analysis analysis NOUN cana-1458 172 18 and and CCONJ cana-1458 172 19 applications application NOUN cana-1458 172 20 , , PUNCT cana-1458 172 21 74 74 NUM cana-1458 172 22 , , PUNCT cana-1458 172 23 pp pp ADJ cana-1458 172 24 432440 432440 NUM cana-1458 172 25 . . PUNCT cana-1458 173 1 [ [ X cana-1458 173 2 6 6 NUM cana-1458 173 3 ] ] PUNCT cana-1458 173 4 hussain hussain PROPN cana-1458 173 5 s s PROPN cana-1458 173 6 and and CCONJ cana-1458 173 7 ahmad ahmad PROPN cana-1458 173 8 b b PROPN cana-1458 173 9 2011 2011 NUM cana-1458 173 10 , , PUNCT cana-1458 173 11 β€˜ ' PUNCT cana-1458 173 12 some some DET cana-1458 173 13 properties property NOUN cana-1458 173 14 of of ADP cana-1458 173 15 soft soft ADJ cana-1458 173 16 topological topological ADJ cana-1458 173 17 spaces space NOUN cana-1458 173 18 ’ ' PUNCT cana-1458 173 19 , , PUNCT cana-1458 173 20 computers computer NOUN cana-1458 173 21 and and CCONJ cana-1458 173 22 mathematics mathematic NOUN cana-1458 173 23 with with ADP cana-1458 173 24 applications application NOUN cana-1458 173 25 , , PUNCT cana-1458 173 26 62 62 NUM cana-1458 173 27 , , PUNCT cana-1458 173 28 pp pp ADP cana-1458 173 29 4058 4058 NUM cana-1458 173 30 - - SYM cana-1458 173 31 4067 4067 NUM cana-1458 173 32 . . PUNCT cana-1458 174 1 [ [ X cana-1458 174 2 7 7 X cana-1458 174 3 ] ] X cana-1458 174 4 japhia japhia PROPN cana-1458 174 5 tino tino PROPN cana-1458 174 6 mercy mercy PROPN cana-1458 174 7 k k PROPN cana-1458 174 8 , , PUNCT cana-1458 174 9 bageerathi bageerathi PROPN cana-1458 174 10 k k PROPN cana-1458 174 11 2024 2024 NUM cana-1458 174 12 , , PUNCT cana-1458 174 13 β€˜ ' PUNCT cana-1458 174 14 a a DET cana-1458 174 15 note note NOUN cana-1458 174 16 on on ADP cana-1458 174 17 fuzzy fuzzy ADJ cana-1458 174 18 soft soft ADJ cana-1458 174 19 sequential sequential ADJ cana-1458 174 20 sets set NOUN cana-1458 174 21 ’ ' PUNCT cana-1458 174 22 , , PUNCT cana-1458 174 23 proceedings proceeding NOUN cana-1458 174 24 of of ADP cana-1458 174 25 national national ADJ cana-1458 174 26 seminar seminar NOUN cana-1458 174 27 on on ADP cana-1458 174 28 β€œ " PUNCT cana-1458 174 29 current current ADJ cana-1458 174 30 scenario scenario NOUN cana-1458 174 31 in in ADP cana-1458 174 32 pure pure ADJ cana-1458 174 33 and and CCONJ cana-1458 174 34 applied applied ADJ cana-1458 174 35 mathematics mathematic NOUN cana-1458 174 36 ” " PUNCT cana-1458 174 37 . . PUNCT cana-1458 175 1 [ [ X cana-1458 175 2 8 8 NUM cana-1458 175 3 ] ] X cana-1458 175 4 mahanta mahanta ADJ cana-1458 175 5 j j PROPN cana-1458 175 6 and and CCONJ cana-1458 175 7 das das PROPN cana-1458 175 8 p p PROPN cana-1458 175 9 k k PROPN cana-1458 175 10 2018 2018 NUM cana-1458 175 11 , , PUNCT cana-1458 175 12 β€˜ ' PUNCT cana-1458 175 13 results result NOUN cana-1458 175 14 on on ADP cana-1458 175 15 fuzzy fuzzy ADJ cana-1458 175 16 soft soft ADJ cana-1458 175 17 topological topological ADJ cana-1458 175 18 spaces space NOUN cana-1458 175 19 ’ ' PUNCT cana-1458 176 1 [ [ X cana-1458 176 2 9 9 NUM cana-1458 176 3 ] ] X cana-1458 176 4 maji maji PROPN cana-1458 176 5 p p PROPN cana-1458 176 6 k k PROPN cana-1458 176 7 , , PUNCT cana-1458 176 8 roy roy PROPN cana-1458 176 9 a a DET cana-1458 176 10 r r NOUN cana-1458 176 11 , , PUNCT cana-1458 176 12 biswas biswas PROPN cana-1458 176 13 r r NOUN cana-1458 176 14 2001 2001 NUM cana-1458 176 15 , , PUNCT cana-1458 176 16 β€˜ ' PUNCT cana-1458 176 17 fuzzy fuzzy ADJ cana-1458 176 18 soft soft ADJ cana-1458 176 19 sets set NOUN cana-1458 176 20 ’ ' PUNCT cana-1458 176 21 , , PUNCT cana-1458 176 22 journal journal NOUN cana-1458 176 23 of of ADP cana-1458 176 24 fuzzy fuzzy ADJ cana-1458 176 25 mathematics mathematic NOUN cana-1458 176 26 , , PUNCT cana-1458 176 27 vol.9 vol.9 PROPN cana-1458 176 28 , , PUNCT cana-1458 176 29 no.3 no.3 VERB cana-1458 176 30 , , PUNCT cana-1458 176 31 pp pp ADV cana-1458 176 32 589 589 NUM cana-1458 176 33 - - SYM cana-1458 176 34 602 602 NUM cana-1458 176 35 . . PUNCT cana-1458 177 1 [ [ X cana-1458 177 2 10 10 NUM cana-1458 177 3 ] ] X cana-1458 177 4 maji maji PROPN cana-1458 177 5 p p PROPN cana-1458 177 6 k k PROPN cana-1458 177 7 , , PUNCT cana-1458 177 8 roy roy PROPN cana-1458 177 9 a a DET cana-1458 177 10 r r NOUN cana-1458 177 11 , , PUNCT cana-1458 177 12 biswas biswas PROPN cana-1458 177 13 r r NOUN cana-1458 177 14 2003 2003 NUM cana-1458 177 15 , , PUNCT cana-1458 177 16 β€˜ ' PUNCT cana-1458 177 17 soft soft ADJ cana-1458 177 18 set set NOUN cana-1458 177 19 theory theory NOUN cana-1458 177 20 ’ ' PUNCT cana-1458 177 21 , , PUNCT cana-1458 177 22 computers computer NOUN cana-1458 177 23 and and CCONJ cana-1458 177 24 mathematics mathematic NOUN cana-1458 177 25 with with ADP cana-1458 177 26 applications application NOUN cana-1458 177 27 , , PUNCT cana-1458 177 28 vol vol NOUN cana-1458 177 29 45 45 NUM cana-1458 177 30 , , PUNCT cana-1458 177 31 pp.555 pp.555 NOUN cana-1458 177 32 - - PUNCT cana-1458 177 33 562 562 NUM cana-1458 177 34 . . PUNCT cana-1458 178 1 [ [ X cana-1458 178 2 11 11 NUM cana-1458 178 3 ] ] SYM cana-1458 178 4 molodstov molodstov NOUN cana-1458 178 5 d d PROPN cana-1458 178 6 1999 1999 NUM cana-1458 178 7 , , PUNCT cana-1458 178 8 β€˜ ' PUNCT cana-1458 178 9 soft soft ADJ cana-1458 178 10 set set ADJ cana-1458 178 11 theory theory NOUN cana-1458 178 12 βˆ’ βˆ’ PROPN cana-1458 178 13 first first ADJ cana-1458 178 14 results result NOUN cana-1458 178 15 ’ ' PUNCT cana-1458 178 16 , , PUNCT cana-1458 178 17 computers computer NOUN cana-1458 178 18 and and CCONJ cana-1458 178 19 mathematics mathematic NOUN cana-1458 178 20 with with ADP cana-1458 178 21 applications application NOUN cana-1458 178 22 , , PUNCT cana-1458 178 23 vol vol NOUN cana-1458 178 24 37 37 NUM cana-1458 178 25 , , PUNCT cana-1458 178 26 pp pp ADV cana-1458 178 27 19 19 NUM cana-1458 178 28 - - SYM cana-1458 178 29 31 31 NUM cana-1458 178 30 . . PUNCT cana-1458 179 1 [ [ X cana-1458 179 2 12 12 NUM cana-1458 179 3 ] ] X cana-1458 179 4 roy roy PROPN cana-1458 179 5 s s PROPN cana-1458 179 6 and and CCONJ cana-1458 179 7 samanta samanta PROPN cana-1458 179 8 t t PROPN cana-1458 179 9 k k PROPN cana-1458 179 10 2012 2012 NUM cana-1458 179 11 , , PUNCT cana-1458 179 12 β€˜ ' PUNCT cana-1458 179 13 a a DET cana-1458 179 14 note note NOUN cana-1458 179 15 on on ADP cana-1458 179 16 fuzzy fuzzy ADJ cana-1458 179 17 soft soft ADJ cana-1458 179 18 topological topological ADJ cana-1458 179 19 spaces space NOUN cana-1458 179 20 ’ ' PUNCT cana-1458 179 21 , , PUNCT cana-1458 179 22 ann.fuzzy ann.fuzzy PUNCT cana-1458 179 23 math.inform math.inform ADJ cana-1458 179 24 , , PUNCT cana-1458 179 25 vol.3 vol.3 PROPN cana-1458 179 26 , , PUNCT cana-1458 179 27 no.2 no.2 PROPN cana-1458 179 28 pp pp NOUN cana-1458 179 29 . . PUNCT cana-1458 180 1 305 305 NUM cana-1458 180 2 - - SYM cana-1458 180 3 311 311 NUM cana-1458 180 4 . . PUNCT cana-1458 181 1 [ [ X cana-1458 181 2 13 13 NUM cana-1458 181 3 ] ] X cana-1458 181 4 serkan serkan PROPN cana-1458 181 5 atmaca atmaca PROPN cana-1458 181 6 , , PUNCT cana-1458 181 7 idris idris PROPN cana-1458 181 8 zorlutuna zorlutuna PROPN cana-1458 181 9 2013 2013 NUM cana-1458 181 10 , , PUNCT cana-1458 181 11 β€˜ ' PUNCT cana-1458 181 12 on on ADP cana-1458 181 13 fuzzy fuzzy ADJ cana-1458 181 14 soft soft ADJ cana-1458 181 15 topological topological ADJ cana-1458 181 16 spaces space NOUN cana-1458 181 17 ’ ' PUNCT cana-1458 181 18 , , PUNCT cana-1458 181 19 annals annal NOUN cana-1458 181 20 of of ADP cana-1458 181 21 fuzzy fuzzy ADJ cana-1458 181 22 mathematics mathematic NOUN cana-1458 181 23 and and CCONJ cana-1458 181 24 informatics informatic NOUN cana-1458 181 25 , , PUNCT cana-1458 181 26 vol.5 vol.5 PROPN cana-1458 181 27 , , PUNCT cana-1458 181 28 no.2 no.2 PROPN cana-1458 181 29 , , PUNCT cana-1458 181 30 pp.377 pp.377 NOUN cana-1458 181 31 - - PUNCT cana-1458 181 32 386 386 NUM cana-1458 181 33 . . PUNCT cana-1458 182 1 [ [ X cana-1458 182 2 14 14 NUM cana-1458 182 3 ] ] X cana-1458 182 4 shabir shabir PROPN cana-1458 182 5 m m PROPN cana-1458 182 6 , , PUNCT cana-1458 182 7 naz naz PROPN cana-1458 182 8 m m PROPN cana-1458 182 9 2011 2011 NUM cana-1458 182 10 , , PUNCT cana-1458 182 11 β€˜ ' PUNCT cana-1458 182 12 on on ADP cana-1458 182 13 soft soft ADJ cana-1458 182 14 topological topological ADJ cana-1458 182 15 spaces space NOUN cana-1458 182 16 ’ ' PUNCT cana-1458 182 17 , , PUNCT cana-1458 182 18 computers computer NOUN cana-1458 182 19 and and CCONJ cana-1458 182 20 mathematics mathematic NOUN cana-1458 182 21 with with ADP cana-1458 182 22 applications application NOUN cana-1458 182 23 , , PUNCT cana-1458 182 24 61(7 61(7 PROPN cana-1458 182 25 ) ) PUNCT cana-1458 182 26 , , PUNCT cana-1458 182 27 pp pp ADP cana-1458 182 28 1786 1786 NUM cana-1458 182 29 - - SYM cana-1458 182 30 1799 1799 NUM cana-1458 182 31 . . PUNCT cana-1458 183 1 [ [ X cana-1458 183 2 15 15 NUM cana-1458 183 3 ] ] X cana-1458 183 4 singha singha PROPN cana-1458 183 5 m m PROPN cana-1458 183 6 , , PUNCT cana-1458 183 7 tamang tamang PROPN cana-1458 183 8 n n PROPN cana-1458 183 9 and and CCONJ cana-1458 183 10 de de PROPN cana-1458 183 11 sarkar sarkar PROPN cana-1458 183 12 s s PROPN cana-1458 183 13 2012 2012 NUM cana-1458 183 14 , , PUNCT cana-1458 183 15 β€˜ ' PUNCT cana-1458 183 16 fuzzy fuzzy ADJ cana-1458 183 17 sequential sequential ADJ cana-1458 183 18 topological topological ADJ cana-1458 183 19 spaces’,arxiv:1209.5835vl[math.gn spaces’,arxiv:1209.5835vl[math.gn NOUN cana-1458 183 20 ] ] X cana-1458 184 1 [ [ X cana-1458 184 2 16 16 NUM cana-1458 184 3 ] ] X cana-1458 184 4 tanay tanay PROPN cana-1458 184 5 b b PROPN cana-1458 184 6 , , PUNCT cana-1458 184 7 kandemir kandemir PROPN cana-1458 184 8 m m PROPN cana-1458 184 9 b b PROPN cana-1458 184 10 2011 2011 NUM cana-1458 184 11 , , PUNCT cana-1458 184 12 β€˜ ' PUNCT cana-1458 184 13 topological topological ADJ cana-1458 184 14 structure structure NOUN cana-1458 184 15 of of ADP cana-1458 184 16 fuzzy fuzzy ADJ cana-1458 184 17 soft soft ADJ cana-1458 184 18 sets set NOUN cana-1458 184 19 ’ ' PUNCT cana-1458 184 20 , , PUNCT cana-1458 184 21 computers computer NOUN cana-1458 184 22 and and CCONJ cana-1458 184 23 mathematics mathematic NOUN cana-1458 184 24 with with ADP cana-1458 184 25 applications application NOUN cana-1458 184 26 , , PUNCT cana-1458 184 27 vol vol NOUN cana-1458 184 28 61 61 NUM cana-1458 184 29 , , PUNCT cana-1458 184 30 pp pp ADP cana-1458 184 31 2952 2952 NUM cana-1458 184 32 - - SYM cana-1458 184 33 2957 2957 NUM cana-1458 184 34 . . PUNCT cana-1458 185 1 [ [ X cana-1458 185 2 17 17 NUM cana-1458 185 3 ] ] X cana-1458 185 4 wong wong PROPN cana-1458 185 5 c c PROPN cana-1458 185 6 k k PROPN cana-1458 185 7 1973 1973 NUM cana-1458 185 8 , , PUNCT cana-1458 185 9 β€˜ ' PUNCT cana-1458 185 10 covering cover VERB cana-1458 185 11 properties property NOUN cana-1458 185 12 of of ADP cana-1458 185 13 fuzzy fuzzy ADJ cana-1458 185 14 topological topological ADJ cana-1458 185 15 spaces space NOUN cana-1458 185 16 ’ ' PUNCT cana-1458 185 17 , , PUNCT cana-1458 185 18 journal journal NOUN cana-1458 185 19 of of ADP cana-1458 185 20 mathematical mathematical ADJ cana-1458 185 21 analysis analysis NOUN cana-1458 185 22 and and CCONJ cana-1458 185 23 applications application NOUN cana-1458 185 24 , , PUNCT cana-1458 185 25 vol.43 vol.43 NOUN cana-1458 185 26 , , PUNCT cana-1458 185 27 pp.697 pp.697 NOUN cana-1458 185 28 - - PUNCT cana-1458 185 29 704 704 NUM cana-1458 185 30 . . PUNCT cana-1458 186 1 [ [ X cana-1458 186 2 18 18 NUM cana-1458 186 3 ] ] PUNCT cana-1458 186 4 zadeh zadeh PROPN cana-1458 186 5 l l PROPN cana-1458 186 6 a a DET cana-1458 186 7 1965 1965 NUM cana-1458 186 8 , , PUNCT cana-1458 186 9 β€˜ ' PUNCT cana-1458 186 10 fuzzy fuzzy ADJ cana-1458 186 11 sets set NOUN cana-1458 186 12 ’ ' PUNCT cana-1458 186 13 , , PUNCT cana-1458 186 14 information information NOUN cana-1458 186 15 and and CCONJ cana-1458 186 16 control control NOUN cana-1458 186 17 , , PUNCT cana-1458 186 18 vol vol NOUN cana-1458 186 19 8 8 NUM cana-1458 186 20 , , PUNCT cana-1458 186 21 pp pp ADV cana-1458 186 22 338 338 NUM cana-1458 186 23 - - SYM cana-1458 186 24 353 353 NUM cana-1458 186 25 . . PUNCT